Distributed resource allocation

ABSTRACT

Disclosed methods, systems, and apparatus, include computer programs encoded on computer storage media, for performing allocation of M resources among N users into K pools by solving a knapsack problem (KP) using a distributed computing system. The method includes: receiving data representing K global constraints and L local constraints of the KP; transforming the KP into a dual problem using K dual multipliers; decomposing the dual problem into N sub-problems; performing two or more iterations in solving the dual problem, wherein in one iteration, for each dual multiplier corresponding to a global constraint corresponding to a pool: determining an updated dual multiplier for the global constraint corresponding to the pool to be a non-negative threshold; and computing M decision variables of each of the N users corresponding to the updated dual multiplier in solving each of the N sub-problems corresponding to the each of the N users.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of and claims the benefit of U.S.patent application Ser. No. 16/915,627, filed Jun. 29, 2020, which is acontinuation of and claims the benefit of U.S. patent application Ser.No. 16/671,069, filed Oct. 31, 2019, which is a continuation of PCTApplication No. PCT/US2019/028973, filed on Apr. 24, 2019, which ishereby incorporated by reference in its entirety.

TECHNICAL FIELD

This specification relates to resource allocation.

BACKGROUND

Knapsack problems (KPs) can often be used to model resource allocationapplications in real-world decision making processes. Examples of KPsinclude asset management, portfolio selection, and budget allocation, toname a few. For example, a KP is solved in order to decide notificationvolume for each user to optimize long-term user engagements. Real-worldresource allocation applications are often approached by solving KPs butthey have been tractable only at a relatively small scale. A number ofvariants of the knapsack problems have been studied in the literature,such as multi-dimensional knapsack problems (MDKPs), multi-choiceknapsack problems (MCKPs), and multi-dimensional multi-choice knapsackproblems (MMKPs). In MDKPs, there are multiple knapsack constraints anda resource item (also referred to as an item) which, when chosen, willconsume resources from multiple knapsacks. MCKP is an extension of theclassical single-constraint KP, where the items are partitioned intomultiple groups and exactly one item from each group can be chosen. MMKPis a combination of MDKP and MCKP.

As one well-known special case of the integer programming (IP) problem,KP (including vanilla KP and its variants such as MDKP, MCKP, and MMKP)is NP-hard. Both exact and heuristic algorithms have been studied forsolving these problems. Existing works studied KPs only at a relativelysmall scale (i.e., thousands to millions of decision variables).

With the increasing number of users, online platforms or serviceproviders (e.g., e-commerce platform, social network platform, andonline financial platform) have an increasing need to solve KPs at alarge scale, for example, in the order of billions or more decisionvariables. Existing resource allocation solutions perform poorly on suchlarge scales as they consume excessive computational resources and time.Techniques for solve KPs at large scale efficiently are desirable.

SUMMARY

This specification describes technologies for resource allocation. Thesetechnologies generally involve solving a knapsack problem (KP) subjectto multiple global constraints and local constraints. The describedtechniques can reduce the computational complexity and improve theconvergence speed in solving the KPs in an iterative manner.

This specification also provides one or more non-transitorycomputer-readable storage media coupled to one or more processors andhaving instructions stored thereon which, when executed by the one ormore processors, cause the one or more processors to perform operationsin accordance with embodiments of the methods provided herein.

This specification further provides a system for implementing themethods provided herein. The system includes one or more processors, anda computer-readable storage medium coupled to the one or more processorshaving instructions stored thereon which, when executed by the one ormore processors, cause the one or more processors to perform operationsin accordance with embodiments of the methods provided herein.

It is appreciated that methods in accordance with this specification mayinclude any combination of the aspects and features described herein.That is, methods in accordance with this specification are not limitedto the combinations of aspects and features specifically describedherein, but also include any combination of the aspects and featuresprovided.

The details of one or more embodiments of this specification are setforth in the accompanying drawings and the description below. Otherfeatures and advantages of this specification will be apparent from thedescription and drawings, and from the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram illustrating an example of a distributed computingsystem for solving a knapsack problem (KP) with an information flow, inaccordance with embodiments of this specification.

FIG. 2 is a pseudocode of an example of a MapReduce algorithm (alsoreferred to as Algorithm 1) for solving knapsack problems, in accordancewith embodiments of this specification.

FIG. 3 is a diagram illustrating an example of a directed acyclic graph(DAG) of resource index sets for local constraints in (15) to (17),respectively, in accordance with embodiments of this specification.

FIG. 4 is a pseudocode of an example of a greedy algorithm (referred toas Algorithm 2) for solving an integer programming (IP) problem subjectto constraints with a hierarchical structure, in accordance withembodiments of this specification.

FIG. 5 is a plot illustrating examples of candidates of a dualmultipliers λ₃ for user i, in accordance with embodiments of thisspecification.

FIG. 6 is a pseudocode of an example of an algorithm for computingcandidate dual multipliers (also referred to as Algorithm 3), inaccordance with embodiments of this specification.

FIG. 7 is a pseudocode of an example of a synchronous coordinate descent(SCD) algorithm (also referred to as Algorithm 4) with a MapReduce modelfor solving a KP, in accordance with embodiments of this specification.

FIG. 8 is a pseudocode of an example of a Map function (also referred toas Algorithm 5) for choosing up to a maximum number (denoted by Q) ofresources for each user, in accordance with embodiments of thisspecification.

FIG. 9 is a pseudocode of an example of a pre-solving by samplingalgorithm (also referred to as Algorithm 6), in accordance withembodiments of this specification.

FIG. 10 is a plot showing examples of optimality gaps between KPsolution using the described algorithm in FIG. 7 and upper boundscomputed by linear programming (LP) relaxation, in accordance withembodiments of this specification.

FIG. 11 is a table (also referred to as Table 1) illustrating examplesof number of SCD iterations with and without pre-solving, in accordancewith embodiments of this specification.

FIG. 12 is a table (also referred to as Table 2) illustrating examplesof experiment results of a distributed system using a SCD algorithm onlarge-scale test data sets, in accordance with embodiments of thisspecification.

FIG. 13 is a plot illustrating an example of an execution result of theMap function 710 and Reduce function 720 for solving a KP, in accordancewith embodiments of this specification.

FIG. 14A is a flowchart of an example of a process for performingresource allocation of M resources among N users into K pools (e.g., Kknapsacks) by solving a knapsack problem (KP), in accordance withembodiments of this specification.

FIG. 14B is a flowchart of an example of a process for performingresource allocation of M resources subject to L constraints, inaccordance with embodiments of this specification.

FIG. 14C is a flowchart of another example of a process for performingresource allocation of M resources among N users into K pools (e.g., Kknapsacks) by solving a knapsack problem (KP), in accordance withembodiments of this specification.

FIG. 15 depicts a block diagram illustrating an example of acomputer-implemented system used to provide computationalfunctionalities associated with described algorithms, methods,functions, processes, flows, and procedures in accordance withembodiments of this specification.

FIG. 16A-C are diagrams of examples of modules of apparatuses inaccordance with embodiments of this specification.

Like reference numbers and designations in the various drawings indicatelike elements.

DETAILED DESCRIPTION

This specification describes technologies for resource allocation. Thesetechnologies generally involve solving large-scale knapsack problems(KPs) in a scalable distributed paradigm via synchronous coordinatedescent (SCD). In some embodiments, the described techniques can helpsolve KPs at the scale of billions or larger with more than one globalconstraint. In some embodiments, the described techniques can help solvesuch large-scale KPs without hurting the optimality of the solution orcompromising the constraint requirements. In some embodiments, thetechnologies can be implemented with off-the-shelf distributed computingframeworks (e.g., MPI, Hadoop, Spark). In some embodiments, thesetechnologies can save memory space, reduce the computational complexity,while improving the solving speed of the KPs. In some embodiments, thesetechnologies can solve resource allocation problems at an unprecedentedscale (e.g., KPs with billions of decisions and constraint variables canbe solved within a few hours (e.g., in the order of O(1) hour) whereasit could be computationally infeasible or significantly time-consumingfor existing techniques to solve KPs on this large of a scale (e.g., KPswith billions of decisions and constraint variables cannot be solved byexisting techniques within a few hours or days).

Knapsack problems (KPs) are very common in real-world decision makingprocesses. Examples of KPs include asset management, portfolioselection, and budget allocation, to name a few. For example, inresource allocation scenarios in a financial industry setting, decisionsare often made on a per user basis while the number of users can belarge, for example, in the order of billion active users.

As an example, an online platform can provide its users a wide spectrumof financial products and services ranging from payment to banking,loans, wealth management, insurances, and so on. Various financialresources may be allocated among users on a daily basis. The resourcescan include, for example, financial budgets (e.g., loans, marketingpromotions, ads spending, asset portfolios) that are to be distributedamong users/user groups, or non-monetary resources such as user traffic(e.g., impressions, clicks, dwell time) that need to be allocated amongdifferent business channels (e.g., HUABEI aka ANT-CREDIT-PAY, JIEBEI akaANT-CASH-NOW, YUEBAO aka ANT-MONEY-MARKET-FUND).

A resource allocation problem can be represented as a KP to optimize ajoint objective, such as the expected user conversions in the case ofmarketing campaign, while respecting a set of constraints. In someembodiments, the constraints can be categorized into global constraintsand local constraints.

Global constraints typically include resource constraints on a globallevel, involving multiple users. For example, a marketing campaign oftenhas a global budget limit as well as maximum amount of resourcesavailable or allowable for each user-cohort (e.g., gender, geo, or userlifecycle stage) or each product channel.

Local constraints include restrictions that are only effective foreither individual users or small groups of users. Throughout thisspecification, the terms “user” and “group” are used interchangeably.For example, a user can include a user group that includes one or moreindividual users. In some embodiments, local constraints are sparse innature. In some embodiments, local constraints demonstrate hierarchicalstructures that can be leveraged to design efficient solvers.

In some embodiments, for a billion-scale KP, both decision variables andlocal constraints are at the level or in the order of billions (i.e.,the order of magnitude is 9, denoted as O(10⁹), or larger), while thesize of the global constraints is often small, for example, in the orderof O(1) to O(100) (i.e., the order of magnitude is 1 to 2). Existingoptimization approaches for solving KPs are only tractable at a muchsmaller scale.

This specification describes technologies for solving real-world KPs atbillion-scale. Using the MapReduce computing model as an example, thisspecification describes a distributed framework for solving KPs byexploiting the decomposability of the dual problem. Instead ofconventional dual descent algorithm that does not work so well as itrequires manual hyper-parameter tuning and is prone to constraintviolations, this specification describes a synchronous coordinatedescent (SCD) algorithm that does not suffer from these issues.

Furthermore, this specification describes a solver for solving aninteger programming (IP) problem subject to constraints that have ahierarchical structures. Unlike off-the-shelf IP solvers such as CPLEXor GUROBI, the solver uses a greedy algorithm that can solve an IPproblem subject to hierarchical structure constraints in polynomialtime. The described IP solver can be referred to as a hierarchicalgreedy IP solver. The hierarchical greedy IP solver can be used in thedistributed framework for solving KPs, for example, by solving an IPsub-problem of the KPs by exploiting the hierarchical structure of thelocal constraints.

Simulation results conducted with both controlled and real-worldsettings show that the distributed SCD algorithm can solve KPs nearlyoptimally and scale nicely to billions of decision variables andconstraints. Moreover, techniques to further speed up the algorithm in adistributed environment are described.

Consider the following generalized variant of a KP:

$\begin{matrix}{{\max\limits_{x_{i,j}}{\sum_{i = 1}^{N}{\sum_{j = 1}^{M}{p_{i,j}x_{i,j}}}}}{{subject}\mspace{14mu}{to}\mspace{14mu}\left( {s.t.} \right)}} & (1) \\{{{\sum_{i = 1}^{N}{\sum_{j = 1}^{M}{b_{i,j,k}x_{i,j}}}} \leq B_{k}},{\forall{k \in \left\{ {1,\ldots\mspace{14mu},K} \right\}}}} & (2) \\{{{\sum_{j = 1}^{M}{c_{i,j,l}x_{i,j}}} \leq C_{l}},{\forall{i \in \left\{ {1,\ldots\mspace{14mu},N} \right\}}},{\forall{l \in \left\{ {1,\ldots\mspace{14mu},L} \right\}}}} & (3) \\{{x_{i,j} \in \left\{ {0,1} \right\}},{\forall{i \in \left\{ {1,\ldots\mspace{14mu},N} \right\}}},{\forall{j \in \left\{ {1,\ldots\mspace{14mu},M} \right\}}}} & (4)\end{matrix}$

where a set of M resources (or items) are to be allocated among a set ofN users respecting K global constraints and L local constraints. Ifresource j is chosen for user i, i.e. x_(i,j)=1, a reward (can also bereferred to as profit, utility, or gain) of p_(i,j) is gained and anamount of resources b_(i,j,k) is consumed for the k-th knapsack, foreach k=1, . . . , K. The input parameters B_(k) and C_(l) are strictlypositive, (i.e., B_(k)>0 and C_(l)>0), while p_(i,j) and b_(i,j,k) arenon-negative. The global constraints (2) limit the resources allocatedfor each knapsack, whereas the local constraints (3) restrict per-userselections. Without loss of generality, here the local constraints areon a per-user basis, as a local constraint on a sparse user set can bedecomposed to a set of per-user constraints.

If 0s are appropriately padded for the cost coefficients c_(i,j,l) suchthat the cost tensor becomes dense, local constraints (3) will vanishand all the constraints will become global. In this sense, theformulation is a generalized variant of the multi-dimensional knapsackproblems (MDKPs).

Note that, in the formulation, only binary decision variables areconsidered, i.e., x_(i,j)∈{0, 1}, but all the techniques described inthe specification can be easily extended to categorical (i.e.,non-negative integer) decision variables. The described techniquessupport categorical variables.

Define the following additional notations for the rest of thespecification:

p_(i): reward vector for user i,

b_(i): global constraint coefficient matrix for user i,

c_(i): local constraint coefficient matrix for user i,

x_(i): decision variable vector for user i,

x: decision variable matrix for all users.

Given that the number of global constraints is much smaller than that oflocal constraints, a set of dual multipliers λ={λ_(k), k=1, 2, . . . ,K} are introduced for the global constraints. Following the Lagrangiantechniques for dual problem transformation, a pair (x, λ) is searchedfor such that x maximizes

$\begin{matrix}{{\max\limits_{x_{i,j}}{\sum_{i = 1}^{N}{\sum_{j = 1}^{M}{p_{i,j}x_{i,j}}}}} - {\sum_{k - 1}^{K}{\lambda_{k}\left( {{\sum_{i - 1}^{N}{\sum_{j = 1}^{M}{b_{i,j,k}x_{i,j}}}} - B_{k}} \right)}}} & (5) \\{\mspace{79mu}{{{s.t.{\sum_{j = 1}^{M}{c_{i,j,l}x_{i,j}}}} \leq C_{l}},{\forall{i \in \left\{ {1,\ldots\mspace{14mu},N} \right\}}},{\forall{l \in \left\{ {1,\ldots\mspace{14mu},L} \right\}}}}} & (6) \\{\mspace{79mu}{{{x_{i,j} \in \left\{ {0,1} \right\}},{\forall{i \in \left\{ {1,\ldots\mspace{14mu},N} \right\}}},{\forall{j \in \left\{ {1,\ldots\mspace{14mu},M} \right\}}}}\mspace{20mu}{{and}\mspace{14mu}{that}\mspace{14mu} x\mspace{14mu}{and}\mspace{14mu}\lambda\mspace{14mu}{jointly}\mspace{14mu}{satisfy}\mspace{14mu}{the}\mspace{14mu}{optimality}\mspace{14mu}{conditions}}}} & (7) \\{\mspace{79mu}{{{\lambda_{k}\left( {{\sum_{i = 1}^{N}{\sum_{j = 1}^{M}{b_{i,j,k}x_{i,j}}}} - B_{k}} \right)} = 0},{\forall{k \in \left\{ {1,\ldots\mspace{14mu},K} \right\}}}}} & (8) \\{\mspace{79mu}{{{{\sum_{i = 1}^{N}{\sum_{j = 1}^{M}{b_{i,j,k}x_{i,j}}}} - B_{k}} \leq 0},{\forall{k \in \left\{ {1,\ldots,K} \right\}}}}} & (9) \\{\mspace{79mu}{{\lambda_{k} \geq 0},{\forall{k \in \left\{ {1,\ldots\mspace{14mu},K} \right\}}}}} & (10)\end{matrix}$

Intuitively, each multiplier λ_(k) (also referred to as Lagrangianmultipliers, dual Lagrangian multipliers, or simply, multipliers) can beinterpreted economically as the shadow price (or the marginal utility)of the k-th knapsack resource.

The maximization problem (5)-(7) is different from a standard zero-oneIP problem formulation due to the existence of (6). However, it can beshown that the optimality conditions still apply. To see this, one canintroduce an additional set of multipliers μs for the local constraints(one for each constraint in (6)) to obtain an IP formulation for (x, μ)jointly.

Given a set of Lagrangian multipliers, λ, the maximization problem in(5) can be decomposed into independent sub-problems, one for each useri.

$\begin{matrix}{{\max\limits_{x_{i,j}}{\sum_{j = 1}^{M}{p_{i,j}x_{i,j}}}} - {\sum_{k = 1}^{K}{\lambda_{k}{\sum_{j = 1}^{M}{b_{i,j,k}x_{i,j}}}}}} & (11) \\{{{s.t.{\sum_{j = 1}^{M}{c_{i,j,l}x_{i,j}}}} \leq C_{l}},{\forall{l \in \left\{ {1,\ldots\mspace{14mu},L} \right\}}}} & (12) \\{{x_{i,j} \in \left\{ {0,1} \right\}},{\forall{j \in \left\{ {1,\ldots\mspace{14mu},M} \right\}}}} & (13)\end{matrix}$

As the size of λ is relatively small, it is feasible to solve eachsubproblem (11)-(13) by any commercially available IP solvers such asCPLEX or GUROBI on a single machine.

A distributed computing framework can be used for the process ofdecomposing the large-scale KPs into many sub-problems of the same form(11)-(13) followed by solving each sub-problem independently. In someembodiments, the process can be implemented using the MapReduce, MPI,HADOOP, SPARK, or any other distributed computing frameworks or models.In the following, MapReduce semantics are used as an example to describethe algorithm. The described techniques can be used in other distributedcomputing framework or models.

FIG. 1 is a diagram illustrating an example of a distributed computingsystem 100 for solving a KP with an information flow, in accordance withembodiments of this specification. The distributed computing system 100(or the distributed framework 100) for solving KPs uses a MapReducecomputing model as an example. In some other embodiments, thedistributed computing system 100 can be implemented using the MPI,HADOOP, SPARK, or any other distributed computing frameworks or models.

The distributed computing system 100 includes multiple IP solvers (alsoreferred to as mappers) 110, multiple aggregators (also referred to asreducers) 120, a dual multiplier updater (also referred to as a masternode) 130. The IP solvers 110 can be, for example, commerciallyavailable or off-the-shelf IP solvers such as CPLEX or GUROBI or acustomized IP solver such as a hierarchical greedy IP solver, asdescribed in more detail below. The multiple IP solvers are distributedamong multiple data processing apparatuses in the distributed computingsystem 100. For example, each IP solver can be run by a singleprocessing core of a computer. In some embodiments, the multiple IPsolvers are implemented by a computing cluster or a cloud computingsystem. The IP solvers 110 can receive user data 140. For example, inthe subproblem (11(-(13), the user data 140 for user i can includep_(i), b_(i) and c_(i). For example, each IP solver 110 can solve thesubproblem (11)-(13) s.t. to local constraints.

In some embodiments, an iterative algorithm is employed to solve for the(x, λ) pair. In each iteration, the solution x_(i,j) for (11) iscomputed by the IP solvers 110, which then emit K values, Σ_(j=1)^(M)b_(i,j,k)x_(i,j), k=1, . . . K, corresponding to the knapsackresources consumed by user i. The reducers 120 aggregate the totalresources consumed for each knapsack, Σ_(i=1) ^(N)Σ_(j=1)^(M)b_(i,j,k)x_(i,j).

The master node 130 can update Lagrangian multipliers A according to oneor more updating rules or algorithms, for example, based on slacks,B_(k)-Σ_(i=1) ^(N)Σ_(j=1) ^(M)b_(i,j,k)x_(i,j). For example, one way toupdate λ_(k) is to apply the dual descent (DD) algorithm. For example,for the (t+1)-th iteration,

λ_(k) ^(t+1)=max(λ_(k) ^(t) +a(Σ_(i=1) ^(N)Σ_(j=1) ^(M) b _(i,j,k) x_(i,j) −B _(k)), 0)   (14)

where the hyper-parameter α is the learning rate.

In some embodiments, the master node 130 can update Lagrangianmultipliers λ according to a synchronous coordinate descent (SCD)algorithm, as described in more detail below, or other algorithms.

In some embodiments, given the Lagrangian multipliers λ, each of the IPsolvers 110 can re-solve Eqs. (11)-(13) for decision variables {x_(i,j)}for each user i. The decision variables can be returned as decisions forall users 150. For example, x_(i,j) can indicate whether an item j(e.g., a certain coupon from HUABEI) will be offered to user i. In someembodiments, the decision variables {x_(i,j)} can be stored and fetchedto serve live traffic. For example, if user i is in production (e.g.,when user i pays using Alipay APP or visits a certain page of AlipayAPP), the platform (e.g., the Alipay APP) can offer the item j to theuser i according to the computed decision variable x_(i,j) for the useri.

FIG. 2 is a pseudocode 200 of an example of a MapReduce algorithm 250(also referred to as Algorithm 1) for solving knapsack problems, inaccordance with embodiments of this specification. In some embodiments,the MapReduce algorithm 250 can include T iterations. Within eachiteration t, at 252, a mapper can execute a respective Map function(e.g., as defined and annotated as 210) for each user. Each Map functionreturns k values related to the K global constraints or K knapsacks,respectively. In some embodiments, N mappers can execute respective Mapfunctions for respective users in parallel.

At 254, for each of the K knapsacks, the total resource consumed by theN users for each knapsack, Σ_(i=1) ^(N)Σ_(j=1) ^(M)b_(i,j,k)x_(i,j), canbe computed by a Reduce function (e.g., as defined and annotated as220). In some embodiments, K reducers can execute respective Reducefunctions for respective knapsacks in parallel.

At 256, the MapReduce algorithm 250 uses dual descent (DD) as the λupdating rule, for example, according to the techniques described w.r.t.Eq. (14). At 258, the MapReduce algorithm 250 terminates and returns λif λ has converged or after a maximum number of T iterations. In someembodiments, λ can be regarded as converged, for example, if thedifference of λ between two iterations is less than a threshold.

Given λ computed by Algorithm 1 as shown in the pseudocode 200, Eqs.(11)-(13) can be re-solved. The decision variables {x_(i,j)} for theusers can be returned, for example, as decisions for all users 150.

In real-world applications, local constraints often demonstrate nestedhierarchical structures, as resources are often organized as nestedusers (e.g., a taxonomy).

As an illustrative example, consider a simplistic marketing campaignwhere there are 10 offers in total, 5 from platform A and 5 fromplatform B; a user is required to be exposed to no more than 2 offersfrom either platform A or platform B and no more than 3 in total.

The local constraints in this example can be expressed as follows:

Σ_(j=1) ⁵x_(i,j)≤2, ∀i∈{1, . . . , N}  (15)

Σ_(j=1) ¹⁰x_(i,j)≤2, ∀i∈{1, . . . , N}  (16)

Σ_(j=1) ¹⁰x_(i,j)<3, ∀i∈{1, . . . , N}  (17)

In this example, local constraints are structured as a 2-level tree. Inpractice, the structure can be represented as a directed acyclic graph(DAG). Some may present much more complex hierarchies. A DAG is a finitedirected graph with no directed cycles. Equivalently, a DAG is adirected graph that has a topological ordering, a sequence of thevertices such that every edge is directed from earlier to later in thesequence. A topological ordering of a directed graph is a linearordering of its vertices such that for every directed edge uv fromvertex u to vertex v, u comes before v in the ordering. In someembodiments, the vertices of the graph may represent tasks to beperformed, and the edges may represent constraints that one task must beperformed before another. In this case, a topological ordering is avalid sequence for the tasks. A topological ordering is possible if andonly if the graph has no directed cycles, that is, if it is a DAG. AnyDAG has at least one topological ordering. A topological ordering of anyDAG can be constructed by known algorithms.

For local constraint l, denote its resource index set by S(l), that is,S(l)={j: c_(i,j,l)=1}. A knapsack problem is considered to have ahierarchical local constraint structure for user i when the followingconditions hold:

c_(i,j,l)∈{0, 1}, ∀j∈{1, . . . , M}, ∀l∈{1, . . . , L},

and

S(l)∩S(l′)=Ø or S(l)⊆S(l′) or S(l)⊇S(l′), ∀l , l′.

The resource index sets are non-overlapping unless one is a subset ofanother. In some embodiments, a DAG can be constructed for the Mresource index sets such that there is an arc from S(l) to S(l) if andonly if S(l)⊆S(l).

FIG. 3 is a diagram illustrating an example of a directed acyclic graph(DAG) 300 of resource index sets 310, 320, and 330 for local constraintsin (15) to (17), respectively, in accordance with embodiments of thisspecification. The resource index set 310 for local constraint in (15)includes resource indices 1-5; the resource index set 320 for localconstraint in (16) includes resource indices 6-10; and the resourceindex set 330 for local constraint in (17) includes resource indices1-10. As illustrated, the resource index set 310 and the resource indexset 320 have no common index. Each of the resource index set 310 and theresource index set 320 is a subset of the resource index set 330.

FIG. 4 is a pseudocode of an example of a greedy algorithm (referred toas Algorithm 2) 400 for solving an IP problem subject to constraintswith a hierarchical structure, in accordance with embodiments of thisspecification. The example of the greedy algorithm 400 can be used tosolve an IP problem under hierarchical local constraints. For example,the optimization problem (11)-(13) for user i can be solved efficientlyby the greedy algorithm that traverses the DAG in a topological order. Asolver configured to implement the example of the greedy algorithm 400can be referred to as a hierarchical greedy IP solver. For example, ahierarchical greedy IP solver can compute decision variables for eachuser given the dual multipliers λ.

The greedy algorithm 400 initially chooses all the resources at 410, forexample, by marking decision variables of x_(i,j) of all resources as 1.At 420, the resource index j is sorted in a non-increasing order basedon a cost-adjusted reward (which is also the contributing dual value ofx_(i,j)),

{tilde over (p)} _(i,j) =p _(i,j)−Σ_(k=1) ^(K)λ_(k) b _(i,j,k).

At 430, the resource index sets {S(l)} are traversed in a topologicalorder of the DAG, for example, by starting at the lowest level of theDAG. For each S(l), at 440, the previously selected resources (i.e.,resources with corresponding decision variable x_(i,j)=1) are fetched ina non-decreasing order of the cost-adjusted reward until their sumexceeds the local constraint C_(l). In some embodiments, it isequivalent to selecting the C_(l) resources with decision variablex_(i,j)=1 and having the top cost-adjusted reward in the resource indexset S(l). The remaining resources in the non-decreasing order of thecost-adjusted reward are thus left un-selected or unchosen.

At 450, the x_(i,j) values for the unchosen resources within S(l) areall marked as 0 and will not be considered anymore in the followingiterations. This greedy selection process is repeated until all thenodes in the DAG have been traversed. At 460, the decision variablesx_(i,j) are output.

As an example, as shown in FIG. 3, the resource index set 310 for localconstraint in (15) (here C_(l)=2), the top C_(l)=2 resources (say,resources 2 and 4) among the resources 1˜5 sorted in the non-decreasingorder of their respective cost-adjusted reward are selected. Thedecision variables of the non-selected resources (i.e., resources 1, 3,and 5 in this example) in resource index set 310 are marked as 0 andwill not be considered again. Similarly, the resource index set 320 forlocal constraint in (16) (here C₂=2), the top C₂=2 resources (say,resources 6 and 9) among the resources 6˜10 sorted in the non-decreasingorder of their respective cost-adjusted reward are selected. In someembodiments, the decision variables of the non-selected resources (i.e.,resources 7, 8, and 10 in this example) in resource index set 320 aremarked as 0 and will not be considered again.

For the resource index set 330 for local constraint in (17) (here C₃=3)which is on a next level of the hierarchy of the DAG 300, there are fourpreviously resources 2, 4, 6 and 9 with decision variables marked as 1.The top C₃=3 resources (say resources 2, 6 and 9) among the fourpreviously resources 2, 4, 6, and 9 sorted in the non-decreasing orderof the cost-adjusted reward are selected. In some embodiments, thedecision variables of the rest resources (i.e., resource 4 in thisexample) in resource index set 330 is marked as 0. In this example, theDAG 300 has been fully traversed and the top three resources (i.e.,resources 2, 6, and 9) are output as the selected resources.

The greedy algorithm, which has a polynomial time complexity, canoptimally solve the maximization problem (11)-(13). A proof ofoptimality is as follows. Given any other solution (denoted by {tildeover (x)}_(i,j), j=1, . . . , M) that satisfies the constraints (12) and(13) but differs from the greedy solution (denoted by x*_(i,j), j=1, . .. , M), the first node in the topological order of the DAG can beidentified at which the resources chosen are different. Due to thenature of the greedy algorithm, there must exist a pair of resources jand j′ at the node where the adjusted reward of resource j is no lessthan that of resource j′, but x*_(i,j)=1, x*_(i,j), =0, {tilde over(x)}_(i,j)=0, {tilde over (x)}_(i,j), =1. {tilde over (x)} can bemodified by setting {tilde over (x)}_(i,j)=1, {tilde over (x)}_(i,j), =0without decreasing the objective value of (11). All the constraints (12)and (13) are still satisfied, because any later node in the topologicalorder of the DAG contains both j and j′ or neither. In this way, anysolution to the greedy solution can be converted without decreasing theobjective value or violating any constraint. This completes the proof.

In some embodiments, with the decision variables solved, for example, byAlgorithm 2 or any other algorithms for solving an IP problem, asynchronous coordinate descent (SCD) algorithm can be used to update thedual multipliers λ.

In some embodiments, for a given set of multipliers λ, Algorithm 2depends on the relative order of the cost-adjusted rewards, {tilde over(p)}_(i,j), without the need of knowing their actual values. Thisproperty allows an efficient algorithm to update λ_(k) satisfying theconditions (8) to (10), while holding other multipliers (λ_(k)′ withk′≠k) fixed. The algorithm can take advantage of the fact that, amongall non-negative values of new λ_(k), the relative order of {tilde over(p)}_(i,j) can potentially change only at the pair-wise intersectionpoints of the M linear functions ƒ_(J)(λ_(k)) of λ_(k), or at theirintersection points (if any) with the horizontal axis. The M linearfunctions ƒ_(j)(λ_(k)) can be, for example, the cost-adjusted rewards(p_(i,j)−Σ_(k′=1, k′≠k) ^(K)λ_(K), b_(i,j,k),)−λ_(k)b_(i,j,k) for j=1, .. . , M or other variations of objective functions of the dual problemin (11) as a function of λ_(k).

FIG. 5 is a plot 500 illustrating examples of candidates of a dualmultiplier λ₃ for a user i, in accordance with embodiments of thisspecification. Assuming that M=2 and K=3, and an example of a localconstraint is:

x_(i,1)+x_(i,2)≤1.

FIG. 5 shows M=2 linear functions 510 and 520 and three new multipliercandidates {λ₃ ⁽¹⁾, λ₃ ⁽²⁾, λ₃ ⁽³⁾} that could affect the optimizationoutcome of (11) for the user i. In particular, λ₃ ⁽¹⁾ is the value of λ₃at the pair-wise intersection point of the linear functions 510 and 520;λ₃ ⁽²⁾ is the value of λ₃ at the pair-wise intersection point of thelinear function 510 and the horizontal axis; and λ₃ ⁽³⁾ is the value ofλ₃ at the pair-wise intersection point of the linear function 520 andthe horizontal axis.

$\begin{matrix}\left\{ {\begin{matrix}{{x_{i,1} = 0},{x_{i,2} = 0},{{{if}\mspace{14mu}{new}\mspace{14mu}\lambda_{3}} \in \left( {\lambda_{3}^{(3)},{+ \infty}} \right)},} \\{{x_{i,1} = 0},{x_{i,2} = 1},{{{if}\mspace{14mu}{new}\mspace{14mu}\lambda_{3}} \in \left( {\lambda_{3}^{(1)},\lambda_{3}^{(3)}} \right\rbrack},} \\{{x_{i,1} = 1},{x_{i,2} = 0},{{{if}\mspace{14mu}{new}\mspace{14mu}\lambda_{3}} \in \left\lbrack {0,\lambda_{3}^{(1)}} \right\rbrack}}\end{matrix}.} \right. & \;\end{matrix}$

This example also shows that it is not necessarily true that theoptimization outcome of (11)-(13) will change at all intersectionpoints. For example, here the maximizers do not change at λ₃ ⁽²⁾.

FIG. 6 is a pseudocode of an example of an algorithm for computingcandidate dual multipliers (also referred to as Algorithm 3) 600, inaccordance with embodiments of this specification. In some embodiments,the algorithm 600 can be used to compute all candidate new dualmultipliers λ_(k) for user i. Specifically, the algorithm 600 includes afunction CalculateIntersectionPoints 650 that can compute theintersection points of M linear functions of λ_(k) (e.g., the objectivefunctions of the dual problem in (11) as a function of λ_(k)) or theirintersection points (if any) with the horizontal axis. For example, asshown in FIG. 6, a function Intersect(ƒ(λ_(k)), g(λ_(k))) 610 is used toreturn non-negative intersection points of ƒ(λ_(k)) and g(λ_(k))>=0,when ƒ(λ_(k)) and g(λ_(k)) are not parallel. For each of the M linearfunctions of λ_(k), its intersection points with the other (M-1) linearfunctions of λ_(k) and the horizontal axis are computed and saved in aset Λ_(k). Non-duplicate values in the set Λ_(k) can be returned as thecandidate new dual multipliers λ_(k) for user i.

Given all candidates of new λ_(k) for each user, a coordinate descentalgorithm that takes into account all users and computes an updatedλ_(k) satisfying the conditions (8)-(10), while having other multipliersfixed.

FIG. 7 is a pseudocode 700 of an example of a synchronous coordinatedescent (SCD) algorithm (also referred to as Algorithm 4) 750 with aMapReduce model for solving a KP, in accordance with embodiments of thisspecification. The synchronous coordinate descent algorithm 750 caninclude T iterations to execute Map and Reduce steps iteratively. Withineach iteration t, in the Map step at 752, a mapper can execute arespective Map function (e.g., as defined and annotated as 710) for eachuser, for example, to solve the IP sub-problems (14)-(16) independently.In some embodiments, N mappers can execute respective Map functions inparallel. In the Reduce step at 754, every λ_(k) is updated by havingother dual multipliers λ_(k′) (k′≠k) fixed. In some embodiments, Kreducers can execute respective Reduce functions in parallel.

Specifically, according to the Map function 710, for each user i, at712, the mapper obtains candidates of new Lagrangian multipliers λ_(k)according to the function CalculateIntersectionPoints 650 as defined inAlgorithm 3 in FIG. 6. At 714, the mapper sorts the candidates of newλ_(k) in a decreasing order. At 715, the mapper goes through allcandidates of Lagrangian multipliers λ_(k) and calculates decisionvariables {x_(i,j)} by solving Eqs. (11)-(13), for example, according toAlgorithm 2 as described w.r.t. to FIG. 4. In some embodiments, given λ,the decision variables {x_(i,j)} can be calculated by solving Eqs.(11)-(13), for example, according to any suitable algorithms of an IPsolver. At 716, the mapper calculates the amount of resources that wouldbe used in the k-th knapsack if updating λ_(k) to the corresponding newvalue. Typically, the amount of used resources increases as λ_(k)decreases. At 718, the mapper emits only the incremental amount of usedresources as λ_(k) decreases. In some embodiments, an advantage of usingthe incremental amount, rather than the total amount, is to allowsumming over all emitted or returned v₂ with v₂≥v in order to determinethe threshold v as done in Reduce function 720 of FIG. 7. If the totalamount is emitted, there could be more than one v₁ that corresponds tothe same user, and assuming (without loss of generality) that at mostone item is allowed per user, additional booking (that means morestorage and computation) may be needed to make sure that no doublecounting of the consumption by that user.

At 754, for each of the K knapsacks, a Reduce function (e.g., as definedand annotated as 720) aggregates the emitted results for the same key(i.e., the knapsack index k) among all users and calculates the updatedλ_(k) to be the minimal threshold v such that the total resources usedfor the k-th knapsack among all users does not exceed B_(k) at 722. Insome embodiments, all candidates of λ_(k) among all users can be sortedin a non-increasing order. The reducer can compute the total resourcesused for the k-th knapsack among all users for a given λ_(k) ^((s)) bysumming the incremental amounts of used resources corresponding to allcandidates of λ_(k) that are no less than λ_(k) ^((s)). The value of theminimal threshold v can be determined such that total resources used forthe k-th knapsack among all users for any given candidate λ_(k)^((s))>=v does not exceed B_(k), whereas total resources used for thek-th knapsack among all users for any given candidate λ_(k) ^((s))<vexceeds B_(k). In some embodiments, the value of the minimal threshold vmay not be one of the candidates of λ_(k). In some embodiments,interpolation can be performed in finding the minimal threshold vcandidates of λ_(k), for example, according to an example describedw.r.t. FIG. 13 below.

The summation as done in the Reduce function 720 properly calculates thetotal resources used by the users for a chosen threshold v, because themapper emits only the incremental amount of resources used as λ_(k)decreases. In some embodiments, K reducers can execute respective Reducefunctions for respective knapsacks in parallel.

FIG. 13 is a plot 1300 illustrating an example of an execution result ofthe Map function 710 and Reduce function 720 for solving a KP, inaccordance with embodiments of this specification. In the example, N=3users are considered. For each of the 3 users, the Map function 710 canbe executed, for example, in parallel by respective IP solvers. Assumethe k-th global constraint corresponding to the k-th knapsack where k=3and B_(k)=28, each user can solve their respective IP problem and obtaintheir respective candidates of λ₃ according to the Map function 710using the techniques described with respect to FIGS. 5-7. Box 1310 showsexecution results of the Map function 710 for each of the three users.For example, as shown in 1311 for User 1, three candidates of λ₃(denoted as vi in the Map function 710) in a decreasing order are 9, 6,and 3, respectively. The total amounts of resources that would be usedin k-th knapsack given the three candidates of λ₃ are 5, 6, and 7,respectively. According to 718, the Mapper function 710 emits only theincremental amount of used resources (denoted as v₂ in the Map function710) as λ_(k) decreases, which are 5, 6−5=1, and 7−6=1, respectively.The emitted output (k, [v₁, v₂]) for User 1 are thus (3, [9, 5]), (3,[6, 1]), and (3, [3, 1]), respectively.

As shown in 1312 for User 2, three candidates of λ₃ (denoted as vi inthe Map function 710) in a decreasing order are 8, 5, and 2,respectively. The total amounts of resources that would be used in k-thknapsack given the three candidates of λ₃ are 10, 12, and 13,respectively. According to 718, the Map function 710 emits only theincremental amount of used resources (denoted as v₂ in the Map function710) as λ_(k) decreases, which are 10, 12−10 =2, and 13−12=1,respectively. The emitted output (k, [v₁, v₂]) for User 2 are thus (3,[8, 10]), (3, [5, 2]), and (3, [2, 1]), respectively.

As shown in 1313 for User 3, three candidates of λ₃ (denoted as vi inthe Map function 710) in a decreasing order are 7, 4, and 1,respectively. The total amounts of resources that would be used in k-thknapsack given the three candidates of λ₃ are 9, 10, and 11,respectively. According to 718, the Mapper function 710 emits only theincremental amount of used resources (denoted as v₂ in the Map function710) as λ_(k) decreases, which are 9, 10−9=1, and 11−10=1, respectively.The emitted output (k, [v₁, v₂]) for User 3 are thus (3, [7, 9]), (3,[4, 1]), and (3, [1, 1]), respectively.

Box 1320 shows an execution result of the Reduce function 720 for allthe three users for the kth global constraint corresponding to the kthknapsack where k=3. In some embodiments, all the candidates of λ₃ acrossall users can be sorted, for example, in a non-increasing order, asshown as [9, 8, 7, 6, 5, 4, 3, 2, and 1] with their respectiveincremental used resources [5, 10, 9, 1, 2, 1, 1, 1, and 1]. In thisexample, given B_(k)=28, it is determined that the total amount ofresources corresponding to λ₃=4 equals 28. The total amount of resourcescorresponding to λ₃=4 can be computed by summing the respectiveincremental used resources (e.g., [5, 10, 9, 1, 2, 1]) for allcandidates of λ₃ that are larger than or equal to 4 (e.g., [9, 8, 7, 6,5, 4]) together. It can also be determined that the total amount ofresources corresponding to any λ₃<4 would exceed B_(k)=28. For example,the total amount of resources corresponding to λ₃=3 can be 29, which canbe computed by summing the respective incremental used resources (e.g.,[5, 10, 9, 1, 2, 1, 1]) for all candidates of λ₃ that are larger than orequal to 3 (e.g., [9, 8, 7, 6, 5, 4, 3]) together. In this example, theminimal threshold v is determined to be 4. λ₃ can be updated to be v=4.

In some embodiments, if B_(k)=28.5, then interpolation can be performedto determine the minimal threshold v. For example, given the totalamount of resources corresponding to λ₃=4 being 28 and the total amountof resources corresponding to λ₃=3 being 29, the minimal threshold v canbe determined to be 3.5 by interpolating the two values of λ₃ todetermine the minimal threshold v of λ₃ such that a total amount ofresources corresponding to the minimal threshold v of λ₃ does not exceedB_(k)=28.5. In some embodiments, other techniques can be used to computethe minimal threshold v.

Referring back to FIG. 7, the process for updating λ can be repeated fora maximum number of T iterations, and terminates early if λ has reachedconvergence. For example, at 756, the coordinate descent algorithm 750terminates and returns λ if λ has converged or after a maximum number ofT iterations.

Given λ computed by Algorithm 1 as shown in the coordinate descentalgorithm 750, Eqs. (11)-(13) can be re-solved. The decision variables{x_(i,j)} for the users can be returned, for example, as decisions forall users 150.

Although the coordinate descent algorithm 750 uses a synchronouscoordinate descent that updates λ_(k) for all k=1, . . . , Ksimultaneously. Other variants of coordinate descent, such as a cycliccoordinate descent that updates one multiplier at a time, and a blockcoordinate descent that updates multiple multipliers in parallel, arealso applicable. In our system, all the aforementioned algorithms aresupported.

For the special case of K=1, it can be shown that Algorithm 4 willconverge to a solution with a total reward that is at mostmax_(i,j)p_(i,j) less than that of the optimal solution, since thealgorithm essentially produces a rounded integer solution after solvinga fractional knapsack problem. For more general cases, it can be shownempirically that the solution computed by Algorithm 4 is nearly optimalfor the class of problem instances.

The Map function in coordinate descent algorithm 750 has a timecomplexity of O(KM³ log M), given M resources per user and K globalconstraints. This complexity can be significantly reduced in somereal-world cases where there exists a one-to-one mapping between theresources and the knapsacks (i.e., M=K, and b_(i,j,k)=0, ∀j, k), andthere is a single local constraint limiting the maximum number (denotedby Q thereafter) of resources chosen for each user.

For such cases, there is at most one candidate of new λ_(k) thatdetermines whether the k-th resource has a top Q adjusted reward or not.Specifically, if the adjusted reward of resource k is already in top Q,the critical value of new λ_(k) is the one that lowers its adjustedreward to the (Q+1)-th adjusted reward. If the adjusted reward ofresource k is not in top Q, the critical value of new λ_(k) is the onethat increases its adjusted reward to the Q-th adjusted reward.

FIG. 8 is a pseudocode 800 of an example of a Map function for choosingup to a maximum number (denoted by Q) of resources for each user (alsoreferred to as Algorithm 5) 850, in accordance with embodiments of thisspecification. As shown in FIG. 8, p is the threshold deciding whetherthe k-th resource will be chosen for user i. If new λ_(k) is larger than

$\frac{p_{i,k} - \overset{\_}{p}}{b_{i,k,k}},$

the updated adjusted reward of the k-th resource will be below p andthus will not be chosen. On the other hand, a new λ_(k) below

$\frac{p_{i,k} - \overset{\_}{p}}{b_{i,k,k}}$

guarantees that the resulting adjusted reward of the k-th resource isamong top Q across all resources. Thus, Algorithm 5 correctly emits theonly candidate of new λ_(k) (if any) that determines whether the k-thresource has a top Q adjusted reward or not.

Algorithm 5 uses quick_select(array, n), an algorithm that returns then-th largest element of an array of length K in a time complexity ofO(K). The overall time complexity of Algorithm 5 is then O(K),independent of the value of Q.

To further accelerate the convergence of the algorithm when scaling itup to solve billion-scale problems, additional algorithms have beendeveloped, which have proven to be effective.

As an example, fine-tuned bucketing can be used to further speed up theconvergence of the algorithm for solving billion-scale problems. Astraightforward implementation of the Reduce function in Algorithm 4, asshown in FIG. 7, is to sort the emitted results by the value of vi andchoose the minimal threshold v based on the sorted results. One way tospeed up is to bucket the values of v₁ and calculate the sum of v₂ foreach bucket. Then the target bucket that the threshold v falls into canbe identified and the value of v can be approximated, for example, byinterpolating within the bucket.

To improve the accuracy of the above approximation through bucketing andinterpolating, in some embodiments, the bucket size can be designed tobe small around the true value of v and large when the bucket isunlikely to contain v. Unfortunately, the true value of v is unknownbeforehand. Nevertheless, due to the iterative nature of Algorithm 4,the value calculated in the previous iteration, i.e., λ^(t) _(k),provides a reasonable estimate for v=λ^(t) _(k) ⁺¹. Thus an unevenbucketing scheme can be designed such that the bucket is of a smaller orminimal size around λ^(t) _(k) and grows (e.g., exponentially or inanother manner) as it deviates from λ^(t) _(k). As an example, given thevalue calculated in the previous iteration λ^(t) _(k), the bucket idassigned to a candidate λ_(k) is given as:

${{bucket\_ id}\left( \lambda_{k} \right)} = {{{sign}\left( {\lambda_{k} - \lambda_{k}^{t}} \right)}\left\lfloor {\log\frac{{\lambda_{k} - \lambda_{k}^{t}}}{\Delta}} \right\rfloor}$

where Δ is a parameter controlling bucket sizes. Additional or differentuneven bucketing scheme can be used.

Like all other iterative algorithms, starting from a good initializationcan significantly accelerate the convergence of the algorithm. In someembodiments, the starting point of the dual multipliers, λ⁰ in Algorithm4, can be chosen randomly. In some embodiments, the starting point ofthe dual multipliers can be estimated by pre-solving using sampled data.For example, by sampling small sets of random users and adjustingknapsack budgets proportionally, Algorithm 4 can be started with betterinitialization. Experiments show that pre-solving can save up to 40% to75% of iterations for large-scale KPs.

FIG. 9 is a pseudocode 900 of an example of a pre-solving by samplingalgorithm (also referred to as Algorithm 6) 950, in accordance withembodiments of this specification. Additional or different pre-solvingby sampling algorithm can be used in connection with the describedalgorithms for solving KP problems. The Algorithm 6 can include riterations of pre-solving. In some embodiments, the τ iterations ofpre-solving can be done sequentially or in parallel. If runsequentially, the average of the λ values computed before the t-thiterations is used to reduce the running time further. As an example,with hundreds of millions of users, n=10, 000 users can be sampled foreach iteration of pre-solving.

With λ returned by Algorithm 4, the decision variables, x_(i,j), foreach user i can be computed. In some embodiments, the total resourcesused across all users may slightly violate the global constraints (2).In some embodiments, a light-weight post-processing method can be usedto strictly ensure the feasibility of the global constraints andaccelerate the convergence. The post-processing method can be based onthe cost-adjusted user reward quantity, which is actually thecontributing dual value of a given user i,

{tilde over (p)} _(i)=Σ_(j=1) ^(M) p _(i,j) x _(i,j)−Σ_(k=1)^(K)λ_(k)Σ_(j=1) ^(M) b _(i,j,k) x _(i,j).

In particular, the users are sorted in a non-decreasing order of {tildeover (p)}_(i), and an entire user of decision variables x_(i) can bereset as 0 until all global constraints are satisfied. Since thecost-adjusted user reward {tilde over (p)}_(i) somehow measures thebenefit of choosing some resources from user i, removing the resourceswith smaller values of {tilde over (p)}_(i) is considered a sensibleheuristic.

Both synthetic and real-world data have been used to test the variousdescribed algorithms for solving the KPs. For example, in theexperiments, synthetic problems with N number of users and K globalconstraints, unless otherwise stated, p_(i,j) is uniformly distributedin [0, 1]. Two classes of global constraints are considered:

-   dense: b_(i,j,k) is uniformly distributed in [0, 1], and-   sparse: M=K, and b_(i,j,k) is uniformly distributed in [0, 1] if j=k    and 0 otherwise.

The budgets of global constraints are set as modest values that scalewith M, N, and L, which practically ensures tightness of constraints.Parameters for local constraints, c_(i,j,l) and C_(l), are all equal to1.

In the following, optimality ratio is defined as the ratio of the primalobjective value to the relaxed LP objective value; constraint violationratio is defined as the ratio of the excessive budget to the givenbudget for a constraint, and we use the max constraint violation ratioover all the constraints to quantify overall violation.

To measure the quality of the KP solution produced by the describedalgorithm, a set of experiments are conducted to evaluate the optimalityratio between the KP solutions (computed by Algorithm 4 withpost-processing method described above) against a relaxed LP solution.

Practically, it is difficult to find an existing LP solver that cancompute the upper bounds for billion-scale KPs to compare against thedescribed distributed solution. For ease of comparison, the optimalitygap is shown on datasets with modest sizes that can be solved byexisting LP tools. The relaxed LP problem is obtained by replacingbinary constraints x_(i,j)∈{0, 1} with 0≤x_(i,j)≤1, ∀i, ∀j, and issolved using GOOGLE OR-tools.

FIG. 10 is a plot 1000 showing examples of optimality gaps between a KPsolution using the described algorithm in FIG. 7 and upper boundscomputed by LP relaxation, in accordance with embodiments of thisspecification. The KP solution is computed according to Algorithm 4 asshown in FIG. 7 with post-processing method described above. The plot900 shows the optimality ratios for N=1, 000, N=10, 000 and K∈{1, 5, 10,15, 20}. In this experiment, the number of resources per user M is fixedat 10. To increase the diversity of the resources, b_(i,j,k) areuniformly distributed in [0, 1] with a probability of 0.5, and in [0,10] with the remaining probability. The optimality ratios for threescenarios of local constraints have been evaluated and shown in FIG. 9where

-   C=[1] means Σ_(j=1) ^(M)x_(i,j)≤1,-   C=[2] means Σ_(j=1) ^(M)x_(i,j)≤2, and-   C=[2,2,3] corresponds to hierarchical local constraints given    by (15) to (17).

For each value of N and scenario of local constraints, the averageoptimality ratio (across 3 runs) are plotted in FIG. 10 as K varies. Asshown in FIG. 10, the optimality gap decreases as N increases. Theoptimality ratio is above 98.6% for all experiment cases, and above99.8% for N=10, 000 under all scenarios of local constraints. Theresults in FIG. 10 support the intuition that when K«N the KP solutioncomputed by Algorithm 4 is nearly optimal.

When the number of users N is large, pre-solving with sampled users canbe used to generate good starting points for λ. In an experiment, n=10,000 users are sampled for each iteration of pre-solving, and thecomputation time of pre-solving is negligible since n«N. The number ofSCD iterations until λ converges with pre-solving are compared againstthe number of iterations needed if always starting at λ_(k)=1.0, ∀k=1, .. . , K.

FIG. 11 is a table 1100 (also referred to as Table 1) illustratingexamples of number of SCD iterations with and without pre-solving, inaccordance with embodiments of this specification. Table 1 shows thenumber of SCD iterations for sparse problem instances with N=1 million,10 million, 100 million. For each N, M=10 and K=10. The results in Table1 show that pre-solving with sampled users reduced the number of SCDiterations by 40% to 75%.

On the other hand, pre-solving alone is not sufficient for solving theKP problem, as the λ produced by pre-solving may lead to constraintviolations. When applying pre-solved λ to full datasets, it is observedthat the number of global constraint violations are 4, 5, and 3 out of10, for N=1 million, 10 million, 100 million, respectively, and thecorresponding max constraint violation ratio is 2.5%, 4.1%, and 3.0%,respectively. However, the distributed SCD solutions have no violations.It is also worth noting that the primal objective value of thepre-solving solution, even with constraint violations, is always smallerthan the distributed SCD solution.

FIG. 12 is a table 1200 (also referred to as Table 2) illustratingexamples of experiment results of a distributed system using a SCDalgorithm (e.g., Algorithm 4) on large-scale test data sets, inaccordance with embodiments of this specification. The large-scale testdata sets contain a number of sparse problem instances with N=100million users, while the number of resources M in each user varies from1 to 100 and thus the total number of resources considered is up to 10billion. The solution quality and the running time are tested. Table 2shows the number of SCD iterations, primal objective values, and dualitygaps. The duality gaps are much smaller than the primal objectivevalues, indicating that the solutions produced are nearly optimal.Furthermore, no global constraint is violated when Algorithm 4terminates (i.e., all constraints are satisfied without anypost-processing).

The running time of the system satisfied the business need of dailyoptimization and adjusting of the decision variables, for example,running with up to 1,000 CPUs in a shared APACHE HADOOP computinginfrastructure (while the exact number of CPUs used is affected by thecluster load) the optimization for 1 billion decision variables andconstraints was able to converge within about 1 hour of wall-clock time(exact wall-time varies due to hardware and workload).

FIG. 14A is a flowchart of an example of a process 1400 a for performingresource allocation of M resources among N users into K pools (e.g., Kknapsacks) by solving a knapsack problem (KP), in accordance withembodiments of this specification. The KP subject to K globalconstraints and L local constraints, for example, as represented in(1)-(4). Each pool can correspond to a respective knapsack as describedin the KP. The KP can represent one or more real-world scenarios such asresource allocation including traffic routing/management, marketingcampaign, product/service promotion, etc. Algorithms 1 and 4 describedabove with respect to FIGS. 2 and 7 are examples of the process 1400 a.

The process 1400 a can be performed using a distributed computingsystem, such as the distributed computing system 100, which includes anumber of processors (e.g., distrusted among different users or locatedin different places). Each of the processors can function as or supportone or more individual solvers, for example, by leveraging multi-coreprocessing capabilities. A solver can be a computer-implemented solver,for example, implemented by software, hardware, or a combination ofboth. Each individual solver can run independently, in parallel witheach other. In some embodiments, some of the individual solvers canfunction as the IP solvers (also referred to as mappers) 110 that canrun independently, in parallel with each other. In some embodiments, oneor more of the processors can function as aggregators (also referred toas reducers) 120 that can run independently, in parallel with eachother. In some embodiments, one or more of the processors can functionas the dual multiplier updater (also referred to as a master node) 130.In some embodiments, a single processor can be used to function as an IPsolver, an aggregator, and a dual multiplier updater, as long as, thatdifferent individual solvers can run independently, in parallel witheach other and different aggregators can run independently, in parallelwith each other, for example, to leverage parallel processingcapabilities and reduce computation time of solving the KP.

In some embodiments, the distributed computing system can be implementedas a cloud computing system. In some embodiments, the distributedcomputing system can have a MapReduce or other distributed computingframework or model. In some embodiments, the distributed computingsystem can be implemented using common distributed platforms such asMPI, HADOOP and SPARK. In some embodiments, the process 1400 a will bedescribed as being performed by a data processing apparatus such as asystem of one or more computers, located in one or more locations, andprogrammed appropriately in accordance with this specification. Forexample, a computer system 1500 of FIG. 15, appropriately programmed,can perform the process 1400 a.

At 1402 a, data representing the K global constraints and the L localconstraints of the KP is received. In some embodiments, L is in an orderof billions or larger. K is smaller than L. Each of the K globalconstraints restricts a respective maximum per-pool cost of the Mresources across two or more users, for example, as shown in Eq. (2).Each of the L local constraints restricts a per-user selection of the Mresources, for example, as shown in Eq. (3).

At 1404 a, the KP is decomposed into N sub-problems using K dualmultipliers (e.g., λ_(k), k=1, 2, . . . , K), for example, in a manneras shown in Eqs. (8)-(13). Each of the N sub-problems corresponding to arespective one of the N users and subject to the L local constraintswith respect to (w.r.t.) the corresponding user, for example, as shownin Eqs. (14)-(16). In some embodiments, N is in an order of a billion orlarger. Each of the K dual multipliers corresponds to a respective oneof the K global constraints. Given the K dual multipliers, the Nsub-problems can be solved independently respecting the L localconstraints w.r.t. the corresponding user, without considering the Kglobal constraints in solving each of the N sub-problems.

At 1406 a, the number of individual solvers (e.g., the IP solver 110)for solving the N sub-problems is determined. In some embodiments,determining the number of individual solvers for solving the Nsub-problems includes determining the number of individual solvers forsolving the N sub-problems based on a specified computation time forsolving the KP (e.g., a desired computation time for solving the KPgiven the large-scale of the KP that involves up to billions of usersand local constraints). In some embodiments, determining the number ofindividual solvers for solving the N sub-problems includes determiningthe number of individual solvers to be equal to or less than N, thenumber of users. In the case that the number of individual solvers is N,each of the N sub-problems can be solved by a respective individualsolver so that the N sub-problems can be solved by the respective Nindividual solvers independently, in parallel with each other.

At 1408 a, the N sub-problems are distributed among the number ofindividual solvers, for example, by assigning each sub-problem to arespective solver. For example, an individual solver can be allocatedwith one or a few (e.g., in the order of one, ten, hundred, or thousand)of the N sub-problems so as to distribute the computational load andreduce the computation time of solving the overall KP by leveraging theparallel processing capabilities of the multiple individual solvers.

At 1410 a, the KP is solved by the distributed computing system byperforming two or more iterations. In some embodiments, a set of initialvalues of the K dual multipliers can be used for decomposing the KP intoN sub-problems at 1404 a. The initial values of the K dual multiplierscan, for example, be determined to be predetermined or random values, orbe estimated by pre-solving using sampled data as described above. The Kdual multipliers can be updated, and the N sub-problems decomposed basedon the K dual multipliers can thus be updated accordingly. As such, theKP can be solved by solving the N sub-problems in an iterative manner,for example, in a MapReduce manner as shown in MapReduce algorithm 250(also referred to as Algorithm 1) in FIG. 1 or synchronous coordinatedescent (SCD) algorithm (also referred to as Algorithm 4) 750 in FIG. 7.

In one iteration, at 1412 a, each of the N sub-problems is solved by oneof the number of individual solvers independently. In some embodiments,each of the N sub-problems is solved by executing a Map function (e.g.,as shown at 252 or 752) by one of the number of individual solvers ormappers independently (e.g., in parallel with each other). The Mapfunction can be defined as the Map function 210 or 710, or in anothermanner. In some embodiments, solving each of the N sub-problems includescomputing an amount of each of the M resources to be allocated to thecorresponding user of the N user (e.g., the decision variable x_(i,j)).

In some embodiments, solving each of the N sub-problems by one of thenumber of individual solvers independently includes solving each of theN sub-problems by the solver to which the sub-problem was assignedindependently. In some embodiments, solving each of the N sub-problemsby one of the number of individual solvers independently includessolving each of the N sub-problems by one of the number of individualsolvers in parallel. In some embodiments, solving by one of the numberof individual solvers independently includes solving each of the Nsub-problems by a respective one of the N individual solvers inparallel. In some embodiments, each of the N sub-problems is an integerprogramming (IP) problem. Accordingly, solving each of the Nsub-problems by one of the number of individual solvers independentlyincludes solving an IP problem by an IP solver independently.

In some embodiments, solving each of the N sub-problems further includescomputing, for each of the K pools, a per-pool cost of the M resourcesby the corresponding user based on the amount of each of the M resourcesto be allocated to the corresponding user, for example, as defined inthe Map function 210. In some embodiments, for each of the K pools, aper-pool cost of the M resources across the N users is computed based onthe amount of each of the M resources to be allocated to thecorresponding user of the N user.

At 1414 a, the K dual multipliers are updated based on the amount ofeach of the M resources to be allocated to the corresponding user of theN user. In some embodiments, the K dual multipliers can be updated byexecuting a Reduce function (e.g., as shown at 254 or 754) by one ormore individual aggregators or reducers. In some embodiments, each ofthe K dual multipliers can be updated by executing a Reduce function(e.g., as shown at 254 or 754) by one of the number of individualaggregators or reducers independently (e.g., in parallel with eachother). The Reduce function can be defined as the Reduce function 220 or720, or in another manner. In some embodiments, the K dual multiplierscan be updated according to dual descent (DD), synchronous coordinatedescent, cyclic coordinate descent, block coordinate descent, or othermultiplier updating algorithm.

In some embodiments, updating the K dual multipliers based on the amountof each of the M resources to be allocated to the corresponding user ofthe N user includes, for example, as the Reduce function 220, updatingeach of the K dual multipliers w.r.t. a corresponding pool based on adifference between a maximum per-pool cost of the M resources across twoor more users for the corresponding pool restricted by a correspondingglobal constraint and a per-pool cost of the M resources across the Nusers for the corresponding pool computed based on the amount of each ofthe M resources to be allocated to the corresponding user of the N user.

In some embodiments, the number of individual reducers for updating theK dual multipliers can be determined, for example, based on a specifiedcomputation time for solving the KP (e.g., a desired computation timefor solving the KP). In some embodiments, determining the number ofindividual reducers for updating the K dual multipliers includesdetermining the number of individual reducers to be equal to or lessthan K, the number of global constraints. In the case where the numberof individual reducers is K, each of the K dual multipliers can beupdated by a respective individual reducer so that the K dualmultipliers can be updated by the respective K individual reducersindependently, in parallel with each other.

At 1416 a, determining whether a convergence condition is met. Theconvergence condition can be defined, for example, based on a maximumnumber of iterations, the K dual multipliers, or both. For example, theconvergence condition can be met if either the K dual multipliers haveconverged or a maximum number of iterations have been executed.Additional or different convergence conditions can be specified.

At 1418 a, in response to determining that a convergence condition ismet (e.g., based on the K dual multipliers), each of the N sub-problemsbased on the K dual multipliers can be solved again by one of the numberof individual solvers independently, wherein solving each of the Nsub-problems includes computing an amount of each of the M resources tobe allocated to the corresponding user of the N user.

At 1420 a, the amount of each of the M resources can be returned as theoutput of the process 1400 a and can be allocated to the correspondinguser of the N user.

FIG. 14B is a flowchart of an example of a process 1400 b for performingresource allocation of M resources subject to L constraints, inaccordance with embodiments of this specification. In some embodiments,the resource allocation of M resources subject to L constraints can berepresented by an integer programming (IP) problem, such as the IPproblem represented by Eqs. (14)-(16). The resource allocation of Mresources subject to L constraints can represent one or more real-worldscenarios such as resource allocation in traffic routing/management,marketing campaign, product/service promotion, etc. Algorithm 2described above with respect to FIG. 4 is an example of the process 1400b.

In some embodiments, the resource allocation of M resources subject to Lconstraints can be a stand-alone problem or one of N sub-problems of aKP subject to K global constraints and the L local constraints, forexample, as represented in (1)-(4). In some embodiments, the process1400 b is performed for one of N users performing resource allocation ofthe M resources among the N users into K pools subject to K globalconstraints and the L constraints, wherein each of the L constraintsrestricts a respective maximum number of resources out of the Mresources to be selected by a single user, and each of the K globalconstraints limits resources allocated for each pool across the N users.

The process 1400 b can be performed using an IP solver (e.g., the mapper110), which can be implemented by one or more processors. In someembodiments, the process 1400 b will be described as being performed bya data processing apparatus such as a system of one or more computers,located in one or more locations, and programmed appropriately inaccordance with this specification. For example, a computer system 1500of FIG. 15, appropriately programmed, can perform the process 1400 b.

At 1402 b, data representing L constraints is received. In someembodiments, each of the L constraints corresponds to a subset of Mresources and restricts a respective maximum number C of resources to beselected among the subset of the M resources, for example, as shown inEq. (3). In some embodiments, the L constraints have a hierarchalstructure, for example, as described w.r.t. FIG. 5. For example, anysubset corresponding to a first constraint out of the L constraints hasno common resource with any other subset corresponding to a secondconstraint out of the L constraints, unless the subset corresponding tothe first constraint is a subset of the other subset corresponding tothe second constraint.

In some embodiments, the data representing the L constraints includes Lindex subsets, each of the L index subsets corresponding to a respectiveone of the L constraints and including indexes of a subset of the Mresources corresponding to the respective one of the L constraints. Forexample, a constraint l of the L constraints corresponds to a subsetS_(l) of the M resources and restricts a maximum number C_(l) ofresources to be selected out of the subset S_(l). In some embodiments,the subset S_(l) includes indexes of a subset of the M resourcescorresponding to the constraint l, where l=1, 2, . . . , L.

In some embodiments, the data representing the L constraints includesdata representing each of the L constraints by a node in a directedacyclic graph (DAG) data structure, for example, as described w.r.t.FIG. 5. In some embodiments, an edge of the DAG represents arelationship between two of the L constraints corresponding to two nodesof the edge. The DAG has no directed cycles.

At 1404 b, a topological ordering of the L constraints is determined,wherein the topological ordering defines a traversing order of the Lconstraints. In some embodiments, the topological ordering of the Lconstraints is determined using existing algorithms for determining atopological ordering of a DAG.

At 1406 b, all the M resources are selected as an initial selection, forexample, by setting the decision variables for each the M resources as 1(or any other value), as shown at 410 in Algorithm 2.

At 1408 b, data representing M rewards is obtained, wherein each rewardcorresponds to one of the M resources, respectively. In someembodiments, the M rewards are M cost-adjusted rewards, respectively,wherein each of the M cost-adjusted rewards includes a reward minus acost if one of the M resources is selected.

At 1410 b, the M resources are sorted in a non-increasing orderaccording to the M rewards corresponding to the M resources (e.g., asshown as 420 in Algorithm 2).

At 1412 b, one or more resources are removed from the initial selectionby traversing each constraint in the topological ordering of the Lconstraints, for example, as shown at 420-450 in Algorithm 2. In someembodiments, the removing includes: at 1414 b, for a constraint l of theL constraints corresponding to a subset S_(l) of the M resources andrestricting a maximum number C_(l) of resources to be selected out ofthe subset S_(l), at 1416 b, determining currently selected resources inthe subset S_(l) (e.g., as shown as 440 in Algorithm 2); and at 1418 b,unselecting, among the currently selected resources in the subset S_(l),any resource that is not one of C_(l) resources that have top C_(l)rewards among the currently selected resources in the subset S_(l)(e.g., by updating the decision variables of the unselected resources tobe 0, as shown at 450 in Algorithm 2).

In some embodiments, unselecting, among the currently selected resourcesin the subset S_(l), any resource that is not one of C_(l) resourcesthat have top C_(l) rewards among the currently selected resources inthe subset S_(l) includes unselecting any resource that is not one ofC_(l) resources that have top C_(l) rewards among the currently selectedresources in the subset S_(l) according to the non-increasing order ofthe M resources.

At 1420 b, it is determined whether all the L constraints in thetopological ordering of the L constraints have been traversed. If so,the process 1400 b proceeds to 1422 b. Otherwise, the process 1400 bgoes back to 1414 b to remove one or more resources for a nextconstraint l=l+l in the topological ordering of the L constraint.

At 1422 b, the selected resources are allocated after traversing all theL constraints in the topological ordering of the L constraints, forexample, by returning or outputting the decision variables for each theM resources, as shown at 460 in Algorithm 2.

FIG. 14C is a flowchart of another example of a process 1400 c forperforming resource allocation of M resources among N users into K pools(e.g., K knapsacks) by solving a knapsack problem (KP), in accordancewith embodiments of this specification. The KP subject to K globalconstraints and L local constraints, for example, as represented in(1)-(4). Each pool can correspond to a respective knapsack as describedin the KP. The KP can represent one or more real-world scenarios such asresource allocation including traffic routing/management, marketingcampaign, product/service promotion, etc. Algorithm 4 described abovewith respect to FIG. 7 can be an example of the process 1400 c.

The process 1400 c can be performed using a distributed computingsystem, such as the distributed computing system 100, which includes anumber of processors (e.g., distrusted among different users or locatedin different places). Each of the processors can function as, orsupport, one or more individual solvers, for example, by leveragingmulti-core processing capabilities. Each individual solver can runindependently, in parallel with each other. In some embodiments, some ofthe individual solvers can function as the IP solvers (also referred toas mappers) 110 that can run independently, in parallel with each other.In some embodiments, one or more of the processors can function asaggregators (also referred to as reducers) 120 that can runindependently, in parallel with each other. In some embodiments, one ormore of the processors can function as the dual multiplier updater (alsoreferred to as a master node) 130. In some embodiments, a singleprocessor can be used to function as an IP solver, an aggregator, and adual multiplier updater, as long as that different individual solverscan run independently, in parallel with each other and differentaggregators can run independently, in parallel with each other, forexample, to leverage parallel processing capabilities and reducecomputation time of solving the KP.

In some embodiments, the distributed computing system can be implementedas a cloud computing system. In some embodiments, the distributedcomputing system can have a MapReduce or other distributed computingframework or model. In some embodiments, the distributed computingsystem can be implemented using common distributed platforms such asMPI, HADOOP, and SPARK. In some embodiments, the process 1400 c will bedescribed as being performed by a data processing apparatus such as asystem of one or more computers, located in one or more locations, andprogrammed appropriately, in accordance with this specification. Forexample, a computer system 1500 of FIG. 15, appropriately programmed,can perform the process 1400 c.

At 1402 c, data representing the K global constraints and the L localconstraints of the KP is received. In some embodiments, L is in an orderof billions or larger. K is smaller than L. Each of the K globalconstraints restricts a respective maximum per-pool cost of the Mresources across two or more users, for example, as shown in Eq. (2).Each of the L local constraints restricts a per-user selection of the Mresources, for example, as shown in Eq. (3).

At 1403 c, the KP is transformed into a dual problem using K dualmultipliers, (e.g., λ_(k), k=1, 2, . . . , K), for example, in a manneras shown in Eqs. (8)-(13). Each of the K dual multipliers corresponds toa respective one of the K global constraints.

At 1404 c, the dual problem is decomposed into N sub-problems, each ofthe N sub-problems corresponding to a respective one of the N users andsubject to the L local constraints with respect to (w.r.t.) therespective one of the N users, for example, as shown in Eqs. (14)-(16).In some embodiments, N is in an order of a billion or larger.

In some embodiments, the L constraints have a hierarchal structure, forexample, as described w.r.t. FIG. 5. For example, any subsetcorresponding to a first constraint out of the L constraints has nocommon resource with any other subset corresponding to a secondconstraint out of the L constraints, unless the subset corresponding tothe first constraint is a subset of the other subset corresponding tothe second constraint.

Given the K dual multipliers, the N sub-problems can be solvedindependently respecting the L local constraints w.r.t. thecorresponding user, without considering the K global constraints insolving each of the N sub-problems.

At 1406 c, the number of individual solvers (e.g., the IP solver 110)for solving the N sub-problems is determined. In some embodiments,determining the number of individual solvers for solving the Nsub-problems includes determining the number of individual solvers forsolving the N sub-problems based on a specified computation time forsolving the KP (e.g., a desired computation time for solving the KPgiven the large-scale of the KP that involves up to billions of usersand local constraints). In some embodiments, determining the number ofindividual solvers for solving the N sub-problems includes determiningthe number of individual solvers to be equal to or less than N, thenumber of users. In the case that the number of individual solvers is N,each of the N sub-problems can be solved by a respective individualsolver so that the N sub-problems can be solved by the respective Nindividual solvers independently, in parallel with each other.

At 1408 c, the N sub-problems are distributed among the number ofindividual solvers. For example, an individual solver can be allocatedwith one or a few (e.g., in the order of one, ten, hundred, or thousand)of the N sub-problems so as to distribute the computational load andreduce the computation time of solving the overall KP by leveraging theparallel processing capabilities of the multiple individual solvers.

At 1410 c, the dual problem is solved by the distributed computingsystem by performing two or more iterations. In some embodiments, at1411 c, a set of initial values of the K dual multipliers can bedetermined for decomposing the KP into N sub-problems at 1403 c. Theinitial values of the K dual multipliers can, for example, be determinedto be predetermined or random values, or be estimated by pre-solvingusing sampled data as described above. For example, the initial valuesof the K dual multipliers are computed based on a sampled version of theKP for resource allocation among S users into the K pools subject to Ksampled global constraints and the L local constraints, wherein the Susers are sampled from the N users, and each of the K sampled globalconstraints restricts a respective scaled-down maximum total cost ofresources across the S users for each of the K pools.

The K dual multipliers can be updated, and the N sub-problems decomposedbased on the K dual multipliers can thus be updated accordingly. Assuch, the KP can be solved by solving the N sub-problems in an iterativemanner, for example, in a MapReduce manner as shown in synchronouscoordinate descent (SCD) algorithm (also referred to as Algorithm 4) 750in FIG. 7.

For example, in one iteration, at 1412 c, each of the N sub-problems issolved by one of the number of individual solvers independently. In someembodiments, solving each of the N sub-problems corresponding to theeach of the N users includes computing M decision variables (e.g., thedecision variable x_(i,j)) of each of the N users corresponding to theupdated dual multiplier, wherein each of the M decision variablesindicates whether or not to select a respective one of the M resourcesby the each of the N users.

In some embodiments, each of the N sub-problems is solved by executing aMap function (e.g., as shown at 752) by one of the number of individualsolvers or mappers independently (e.g., in parallel with each other).The Map function can be defined as the Map function 710, or in anothermanner. For example, solving each of the N sub-problems corresponding tothe each of the N users includes: for a sub-problem corresponding to auser and for each dual multiplier corresponding to the global constraintcorresponding to the pool, computing candidates of the dual multiplier;sorting the candidates of the dual multiplier in a non-increasing order;traversing the candidates of the dual multiplier in the non-increasingorder: for each candidate of the dual multiplier, computing M decisionvariables of the user corresponding to the candidate of the dualmultiplier; and computing an incremental cost of selected resourcesbased on the M decision variables of the user corresponding to thecandidate of the dual multiplier relative to a cost of selectedresources based on the M decision variables of the user corresponding toa candidate of the dual multiplier preceding the candidate of the dualmultiplier in the non-increasing order. In some embodiments, a totalcost of resources selected across the N users for the pool correspondingto a certain dual multiplier is computed by summing incremental costs ofselected resources of the user corresponding to all candidates of thedual multiplier that are larger than or equal to the certain dualmultiplier.

In some embodiments, computing candidates of the dual multiplierincludes computing pair-wise intersections points of M linear functionsof the dual multiplier, wherein the M linear functions of the dualmultiplier represent M cost-adjusted rewards of the M resources giventhe dual multiplier.

In some embodiments, the non-negative threshold is computed based on thecandidates of the dual multiplier sorted in the non-increasing order anda respective total cost of resources selected across the N users for thepool corresponding to one of the candidates of the dual multiplier inthe non-increasing order, for example, according to the techniques asdescribed w.r.t. FIG. 13. In some embodiments, interpolation isperformed on the candidates of the dual multiplier sorted in thenon-increasing order for the computing the non-negative threshold, forexample, according to the techniques as described w.r.t. FIG. 13.

In some embodiments, bucketing schemes such as the fine-tuned bucketingalgorithm described above can be performed to speed up the convergenceof the process 1400 c. For example, the candidates of the dualmultiplier are grouped into a plurality of non-uniform budgets; sums ofincremental costs of selected resources of the user corresponding torespective candidates of the dual multiplier in the plurality ofnon-uniform budgets are computed; and the non-negative threshold iscomputed based on the sums of incremental costs.

In some embodiments, solving each of the N sub-problems by one of thenumber of individual solvers independently includes solving each of theN sub-problems by the solver to which the sub-problem was assignedindependently. In some embodiments, solving each of the N sub-problemsby one of the number of individual solvers independently includessolving each of the N sub-problems by one of the number of individualsolvers in parallel. In some embodiments, solving by one of the numberof individual solvers independently includes solving each of the Nsub-problems by a respective one of the N individual solvers inparallel. In some embodiments, each of the N sub-problems is an integerprogramming (IP) problem. Accordingly, solving each of the Nsub-problems by one of the number of individual solvers independentlyincludes solving an IP problem by an IP solver (e.g., the hierarchicalgreedy IP solver as described w.r.t. FIG. 4) independently.

At 1414 c, for each dual multiplier corresponding to a global constraintcorresponding to a pool that restricts a maximum per-pool cost of the Mresources across two or more users, an updated dual multiplier for theglobal constraint corresponding to the pool is determined, for example,to be a non-negative threshold such as the minimal threshold v as shownat 722 of Algorithm 4 in FIG. 7. The non-negative threshold can be suchthat a total cost of resources selected across the N users for the poolcorresponding to a dual multiplier that is less than the non-negativethreshold exceeds the maximum per-pool cost; and a total cost ofresources selected across the N users for the pool corresponding to adual multiplier that is not less than the non-negative threshold doesnot exceed the maximum per-pool cost.

At 1416 c, determining whether a convergence condition is met. Theconvergence condition can be defined, for example, based on a maximumnumber of iterations, the K dual multipliers, or both. For example, theconvergence condition can be met if either the K dual multipliers haveconverged or a maximum number of iterations have been executed.Additional or different convergence conditions can be specified.

At 1418 c, in response to determining that a convergence condition ismet (e.g., based on the K dual multipliers), each of the N sub-problemsbased on the K dual multipliers can be solved again by one of the numberof individual solvers independently, wherein solving each of the Nsub-problems includes computing M decision variables of each of the Nusers corresponding to the K dual multipliers.

At 1420 c, the M resources are allocated among the N users according tothe M decision variables of each user of the N users. The M decisionvariables of each user of the N users can be returned as the output ofthe process 1400 c.

FIG. 15 depicts a block diagram illustrating an example of acomputer-implemented system 1500 used to provide computationalfunctionalities associated with described algorithms, methods,functions, processes, flows, and procedures in accordance withembodiments of this specification. FIG. 15 is a block diagramillustrating an example of a computer-implemented system 1500 used toprovide computational functionalities associated with describedalgorithms, methods, functions, processes, flows, and procedures,according to an embodiment of the present disclosure. In the illustratedembodiment, System 1500 includes a Computer 1502 and a Network 1530.

The illustrated Computer 1502 is intended to encompass any computingdevice such as a server, desktop computer, laptop/notebook computer,wireless data port, smart phone, personal data assistant (PDA), tabletcomputer, one or more processors within these devices, another computingdevice, or a combination of computing devices, including physical orvirtual instances of the computing device, or a combination of physicalor virtual instances of the computing device. Additionally, the Computer1502 can include an input device, such as a keypad, keyboard, touchscreen, another input device, or a combination of input devices that canaccept user information, and an output device that conveys informationassociated with the operation of the Computer 1502, including digitaldata, visual, audio, another type of information, or a combination oftypes of information, on a graphical-type user interface (UI) (or GUI)or other UI.

The Computer 1502 can serve in a role in a distributed computing systemas a client, network component, a server, a database or anotherpersistency, another role, or a combination of roles for performing thesubject matter described in the present disclosure. The illustratedComputer 1502 is communicably coupled with a Network 1530. In someembodiments, one or more components of the Computer 1502 can beconfigured to operate within an environment, includingcloud-computing-based, local, global, another environment, or acombination of environments.

At a high level, the Computer 1502 is an electronic computing deviceoperable to receive, transmit, process, store, or manage data andinformation associated with the described subject matter. According tosome embodiments, the Computer 1502 can also include or be communicablycoupled with a server, including an application server, e-mail server,web server, caching server, streaming data server, another server, or acombination of servers.

The Computer 1502 can receive requests over Network 1530 (for example,from a client software application executing on another Computer 1502)and respond to the received requests by processing the received requestsusing a software application or a combination of software applications.In addition, requests can also be sent to the Computer 1502 frominternal users (for example, from a command console or by anotherinternal access method), external or third-parties, or other entities,individuals, systems, or computers.

Each of the components of the Computer 1502 can communicate using aSystem Bus 1503. In some embodiments, any or all of the components ofthe Computer 1502, including hardware, software, or a combination ofhardware and software, can interface over the System Bus 1503 using anapplication programming interface (API) 1512, a Service Layer 1513, or acombination of the API 1512 and Service Layer 1513. The API 1512 caninclude specifications for routines, data structures, and objectclasses. The API 1512 can be either computer-language independent ordependent and refer to a complete interface, a single function, or evena set of APIs. The Service Layer 1513 provides software services to theComputer 1502 or other components (whether illustrated or not) that arecommunicably coupled to the Computer 1502. The functionality of theComputer 1502 can be accessible for all service consumers using theService Layer 1513. Software services, such as those provided by theService Layer 1513, provide reusable, defined functionalities through adefined interface. For example, the interface can be software written inJAVA, C++, another computing language, or a combination of computinglanguages providing data in extensible markup language (XML) format,another format, or a combination of formats. While illustrated as anintegrated component of the Computer 1502, alternative embodiments canillustrate the API 1512 or the Service Layer 1513 as stand-alonecomponents in relation to other components of the Computer 1502 or othercomponents (whether illustrated or not) that are communicably coupled tothe Computer 1502. Moreover, any or all parts of the API 1512 or theService Layer 1513 can be implemented as a child or a sub-module ofanother software module, enterprise application, or hardware modulewithout departing from the scope of the present disclosure.

The Computer 1502 includes an Interface 1504. Although illustrated as asingle Interface 1504, two or more Interfaces 1504 can be used accordingto particular needs, desires, or particular embodiments of the Computer1502. The Interface 1504 is used by the Computer 1502 for communicatingwith another computing system (whether illustrated or not) that iscommunicatively linked to the Network 1530 in a distributed environment.Generally, the Interface 1504 is operable to communicate with theNetwork 1530 and includes logic encoded in software, hardware, or acombination of software and hardware. More specifically, the Interface1504 can include software supporting one or more communication protocolsassociated with communications such that the Network 1530 or hardware ofInterface 1504 is operable to communicate physical signals within andoutside of the illustrated Computer 1502.

The Computer 1502 includes a Processor 1505. Although illustrated as asingle Processor 1505, two or more Processors 1505 can be used accordingto particular needs, desires, or particular embodiments of the Computer1502. Generally, the Processor 1505 executes instructions andmanipulates data to perform the operations of the Computer 1502 and anyalgorithms, methods, functions, processes, flows, and procedures asdescribed in the present disclosure.

The Computer 1502 also includes a Database 1506 that can hold data forthe Computer 1502, another component communicatively linked to theNetwork 1530 (whether illustrated or not), or a combination of theComputer 1502 and another component. For example, Database 1506 can bean in-memory, conventional, or another type of database storing dataconsistent with the present disclosure. In some embodiments, Database1506 can be a combination of two or more different database types (forexample, a hybrid in-memory and conventional database) according toparticular needs, desires, or particular embodiments of the Computer1502 and the described functionality. Although illustrated as a singleDatabase 1506, two or more databases of similar or differing types canbe used according to particular needs, desires, or particularembodiments of the Computer 1502 and the described functionality. WhileDatabase 1506 is illustrated as an integral component of the Computer1502, in alternative embodiments, Database 1506 can be external to theComputer 1502. As an example, Database 1506 can include theabove-described global constraints 1516 and local constraints 1518 of aKP.

The Computer 1502 also includes a Memory 1507 that can hold data for theComputer 1502, another component or components communicatively linked tothe Network 1530 (whether illustrated or not), or a combination of theComputer 1502 and another component. Memory 1507 can store any dataconsistent with the present disclosure. In some embodiments, Memory 1507can be a combination of two or more different types of memory (forexample, a combination of semiconductor and magnetic storage) accordingto particular needs, desires, or particular embodiments of the Computer1502 and the described functionality. Although illustrated as a singleMemory 1507, two or more Memories 1507 or similar or differing types canbe used according to particular needs, desires, or particularembodiments of the Computer 1502 and the described functionality. WhileMemory 1507 is illustrated as an integral component of the Computer1502, in alternative embodiments, Memory 1507 can be external to theComputer 1502.

The Application 1508 is an algorithmic software engine providingfunctionality according to particular needs, desires, or particularembodiments of the Computer 1502, particularly with respect tofunctionality described in the present disclosure. For example,Application 1508 can serve as one or more components, modules, orapplications. Further, although illustrated as a single Application1508, the Application 1508 can be implemented as multiple Applications1508 on the Computer 1502. In addition, although illustrated as integralto the Computer 1502, in alternative embodiments, the Application 1508can be external to the Computer 1502.

The Computer 1502 can also include a Power Supply 1514. The Power Supply1514 can include a rechargeable or non-rechargeable battery that can beconfigured to be either user- or non-user-replaceable. In someembodiments, the Power Supply 1514 can include power-conversion ormanagement circuits (including recharging, standby, or another powermanagement functionality). In some embodiments, the Power Supply 1514can include a power plug to allow the Computer 1502 to be plugged into awall socket or another power source to, for example, power the Computer1502 or recharge a rechargeable battery.

There can be any number of Computers 1502 associated with, or externalto, a computer system containing Computer 1502, each Computer 1502communicating over Network 1530. Further, the term “client,” “user,” orother appropriate terminology can be used interchangeably, asappropriate, without departing from the scope of the present disclosure.Moreover, the present disclosure contemplates that many users can useone Computer 1502, or that one user can use multiple computers 1502.

FIG. 16A is a diagram of on example of modules of an apparatus 1600 a inaccordance with embodiments of this specification. The apparatus 1600 acan be an example embodiment of a data processing apparatus, such as adistributed computing system that includes a number of individualcomputer-implemented solvers, for performing resource allocation of Mresources among N users into K pools by solving a knapsack problem (KP)subject to K global constraints and L local constraints. The apparatus1600 a can correspond to the embodiments described above, and theapparatus 1600 a includes the following: a receiving module 1601 a forreceiving data representing the K global constraints and the L localconstraints, wherein K is smaller than L, where each of the K globalconstraints restricts a respective maximum per-pool cost of the Mresources across two or more users, and each of the L local constraintsrestricts a per-user selection of the M resources; a decomposing module1602 a for decomposing the KP into N sub-problems using K dualmultipliers, each of the N sub-problems corresponding to a respectiveone of the N users and subject to the L local constraints with respectto (w.r.t.) the corresponding user, wherein N is in an order of billionsor larger, wherein each of the K dual multipliers corresponds to arespective one of the K global constraints; a first determining module1603 a for determining the number of individual computer-implementedsolvers for solving the N sub-problems; a distributing module 1604 a fordistributing the N sub-problems among the number of individualcomputer-implemented solvers; and a first solving module 1605 a forsolving the KP by the distributed computing system by performing two ormore iterations. The first solving module 1605 a further includes afirst solving sub-module for, in one iteration, solving each of the Nsub-problems by one of the number of individual computer-implementedsolvers independently, wherein solving each of the N sub-problemsincludes computing an amount of each of the M resources to be allocatedto the corresponding user of the N user; and an updating sub-module forupdating the K dual multipliers based on the amount of each of the Mresources to be allocated to the corresponding user of the N user.

In an optional embodiment, L is in an order of billions or larger.

In an optional embodiment, determining the number of individualcomputer-implemented solvers for solving the N sub-problems includesdetermining the number of individual computer-implemented solvers forsolving the N sub-problems based on a specified computation time forsolving the KP; and solving each of the N sub-problems by one of thenumber of individual computer-implemented solvers independently includessolving each of the N sub-problems by one of the number of individualcomputer-implemented solvers in parallel.

In an optional embodiment, determining the number of individualcomputer-implemented solvers for solving the N sub-problems includesdetermining the number of individual computer-implemented solvers to beN, and solving each of the N sub-problems by one of the number ofindividual computer-implemented solvers independently includes solvingeach of the N sub-problems by a respective one of the N individualcomputer-implemented solvers in parallel.

In an optional embodiment, solving each of the N sub-problems by one ofthe number of individual computer-implemented solvers independentlyincludes solving an integer programming (IP) problem by an IPcomputer-implemented solver independently.

In an optional embodiment, the apparatus 1600 a further includes: asecond determining module 1606 a for determining whether a convergencecondition is met based on the K dual multipliers; a second solvingmodule 1607 a for solving each of the N sub-problems based on the K dualmultipliers by one of the number of individual computer-implementedsolvers independently in response to determining that a convergencecondition is met based on the K dual multipliers, wherein solving eachof the N sub-problems includes computing an amount of each of the Mresources to be allocated to the corresponding user of the N user, andan allocating module 1608 a for allocating the amount of each of the Mresources to the corresponding user of the N user.

In an optional embodiment, solving each of the N sub-problems furtherincludes computing, for each of the K pools, a per-pool cost of the Mresources by the corresponding user based on the amount of each of the Mresources to be allocated to the corresponding user.

In an optional embodiment, the apparatus 1600 a further includes: acomputing module for computing, in one iteration, for each of the Kpools, a per-pool cost of the M resources across the N users based onthe amount of each of the M resources to be allocated to thecorresponding user of the N users; and the updating sub-module thatupdates each of the K dual multipliers w.r.t. a corresponding pool basedon a difference between a maximum per-pool cost of the M resourcesacross two or more users for the corresponding pool restricted by acorresponding global constraint and a per-pool cost of the M resourcesacross the N users for the corresponding pool computed based on theamount of each of the M resources to be allocated to the correspondinguser of the N user.

The system, apparatus, module, or unit illustrated in the previousembodiments can be implemented by using a computer chip or an entity, orcan be implemented by using a product having a certain function. Atypical embodiment device is a computer, and the computer can be apersonal computer, a laptop computer, a cellular phone, a camera phone,a smartphone, a personal digital assistant, a media player, a navigationdevice, an email receiving and sending device, a game console, a tabletcomputer, a wearable device, or any combination of these devices.

For an embodiment process of functions and roles of each module in theapparatus, references can be made to an embodiment process ofcorresponding steps in the previous method. Details are omitted here forsimplicity.

Because an apparatus embodiment basically corresponds to a methodembodiment, for related parts, references can be made to relateddescriptions in the method embodiment. The previously describedapparatus embodiment is merely an example. The modules described asseparate parts may or may not be physically separate, and partsdisplayed as modules may or may not be physical modules, may be locatedin one position, or may be distributed on a number of network modules.Some or all of the modules can be selected based on actual demands toachieve the objectives of the solutions of the specification. A personof ordinary skill in the art can understand and implement theembodiments of the present application without creative efforts.Referring again to FIG. 16A, it can be interpreted as illustrating aninternal functional module and a structure of a data processingapparatus, such as a distributed computing system that includes a numberof individual computer-implemented solvers, for performing resourceallocation of M resources among N users into K pools by solving aknapsack problem (KP) subject to K global constraints and L localconstraints. An execution body in essence can be an electronic device,and the electronic device includes the following: one or more processorsand a memory configured to store an executable instruction of the one ormore processors.

The one or more processors are configured to receive data representingthe K global constraints and the L local constraints, wherein K issmaller than L, each of the K global constraints restricts a respectivemaximum per-pool cost of the M resources across two or more users, andeach of the L local constraints restricts a per-user selection of the Mresources; decompose the KP into N sub-problems using K dualmultipliers, each of the N sub-problems corresponding to a respectiveone of the N users and subject to the L local constraints with respectto (w.r.t.) the corresponding user, wherein N is in an order of billionor larger, wherein each of the K dual multipliers corresponds to arespective one of the K global constraints; determine the number ofindividual computer-implemented solvers for solving the N sub-problems;distribute the N sub-problems among the number of individualcomputer-implemented solvers; and solve the KP by the distributedcomputing system by performing two or more iterations. In one iteration,the one or more processors are configured to solve each of the Nsub-problems by one of the number of individual computer-implementedsolvers independently, wherein solving each of the N sub-problemsincludes computing an amount of each of the M resources to be allocatedto the corresponding user of the N user; and update the K dualmultipliers based on the amount of each of the M resources to beallocated to the corresponding user of the N user.

Optionally, L is in an order of billions or larger.

Optionally, determining the number of individual computer-implementedsolvers for solving the N sub-problems includes determining the numberof individual computer-implemented solvers for solving the Nsub-problems based on a specified computation time for solving the KPand solving each of the N sub-problems by one of the number ofindividual computer-implemented solvers independently includes solvingeach of the N sub-problems by one of the number of individualcomputer-implemented solvers in parallel.

Optionally, determining the number of individual computer-implementedsolvers for solving the N sub-problems includes determining the numberof individual computer-implemented solvers to be N, and solving each ofthe N sub-problems by one of the number of individualcomputer-implemented solvers independently includes solving each of theN sub-problems by a respective one of the N individualcomputer-implemented solvers in parallel.

Optionally, solving each of the N sub-problems by one of the number ofindividual computer-implemented solvers independently includes solvingan integer programming (IP) problem by an IP computer-implemented solverindependently.

Optionally, the one or more processors are configured to determinewhether a convergence condition is met based on the K dual multipliers;in response to determining that a convergence condition is met based onthe K dual multipliers, solve each of the N sub-problems based on the Kdual multipliers by one of the number of individual computer-implementedsolvers independently, wherein solving each of the N sub-problemsincludes computing an amount of each of the M resources to be allocatedto the corresponding user of the N user; and allocate the amount of eachof the M resources to the corresponding user of the N user.

Optionally, solving each of the N sub-problems further includescomputing, for each of the K pools, a per-pool cost of the M resourcesby the corresponding user based on the amount of each of the M resourcesto be allocated to the corresponding user.

Optionally, the one or more processors are configured to compute foreach of the K pools, a per-pool cost of the M resources across the Nusers based on the amount of each of the M resources to be allocated tothe corresponding user of the N user; and wherein updating the K dualmultipliers based on the amount of each of the M resources to beallocated to the corresponding user of the N user includes updating eachof the K dual multipliers w.r.t. a corresponding pool based on adifference between a maximum per-pool cost of the M resources across twoor more users for the corresponding pool restricted by a correspondingglobal constraint and a per-pool cost of the M resources across the Nusers for the corresponding pool computed based on the amount of each ofthe M resources to be allocated to the corresponding user of the N user.

FIG. 16B is a diagram of on example of modules of an apparatus 1600 b inaccordance with embodiments of this specification. The apparatus 1600 bcan be an example embodiment of a data processing apparatus forallocating M resources subject to L constraints. The apparatus 1600 bcan correspond to the embodiments described above, and the apparatus1600 b includes the following: a receiving module 160 lb for receivingdata representing L constraints, wherein each of the L constraintscorresponds to a subset of M resources and restricts a respectivemaximum number C of resources to be selected among the subset of the Mresources, wherein any subset corresponding to a first constraint out ofthe L constraints has no common resource with any other subsetcorresponding to a second constraint out of the L constraints, unlessthe subset corresponding to the first constraint is a subset of theother subset corresponding to the second constraint; a determiningmodule 1602 b for determining a topological ordering of the Lconstraints, wherein the topological ordering defines a traversing orderof the L constraints; a selecting module 1603 b for selecting all the Mresources as an initial selection; a removing module 1604 b for removingresources from the initial selection by traversing each constraint inthe topological ordering of the L constraints, wherein the removingmodule 1604 b further includes a determining sub-module 1605 b for, fora constraint l of the L constraints corresponding to a subset S_(l) ofthe M resources and restricting a maximum number C_(l) of resources tobe selected out of the subset S_(l), determining currently selectedresources in the subset S_(l); and an unselecting sub-module 1606 b forunselecting, among the currently selected resources in the subset S_(l),any resource that is not one of C_(l) resources that have top C_(l)rewards among the currently selected resources in the subset S_(l); andan allocating module 1607 b for allocating the selected resources aftertraversing all the L constraints in the topological ordering of the Lconstraints.

In an optional embodiment, the data representing the L constraintsincludes L index subsets, each of the L index subsets corresponds to arespective one of the L constraints and includes indexes of a subset ofthe M resources corresponding to the respective one of the Lconstraints; and the subset S_(l) includes indexes of a subset of the Mresources corresponding to the constraint l.

In an optional embodiment, the data representing the L constraintsincludes data representing each of the L constraints by a node in adirected acyclic graph (DAG) data structure, and an edge of the DAGrepresents a relationship between two of the L constraints correspondingto two nodes of the edge.

In an optional embodiment, the apparatus 1600 b further includes: anobtaining module for obtaining data representing M rewards, each rewardcorresponding to one of the M resources respectively.

In an optional embodiment, the apparatus 1600 b further includes: asorting module for sorting the M resources in a non-increasing orderaccording to the M rewards corresponding to the M resources beforetraversing each subset in the L subsets in the topological ordering ofthe L subsets; and wherein the unselecting sub-module is capable ofunselecting the any resource that is not one of C_(l) resources thathave top C_(l) rewards among the currently selected resources in thesubset S_(l) according to the non-increasing order of the M resources.

In an optional embodiment, the M rewards are M cost-adjusted rewardsrespectively, wherein each of the M cost-adjusted rewards includes areward minus a cost if one of the M resources is selected.

In an optional embodiment, the apparatus 1600 b is used for performingfor one of N users in performing resource allocation of the M resourcesamong the N users into K pools subject to K global constraints and the Lconstraints, wherein each of the L constraints restricts a respectivemaximum number of resources out of the M resources to be selected by asingle user, and each of the K global constraints limits resourcesallocated for each pool across the N users.

The system, apparatus, module, or unit illustrated in the previousembodiments can be implemented by using a computer chip or an entity, orcan be implemented by using a product having a certain function. Atypical embodiment device is a computer, and the computer can be apersonal computer, a laptop computer, a cellular phone, a camera phone,a smartphone, a personal digital assistant, a media player, a navigationdevice, an email receiving and sending device, a game console, a tabletcomputer, a wearable device, or any combination of these devices.

For an embodiment process of functions and roles of each module in theapparatus, references can be made to an embodiment process ofcorresponding steps in the previous method. Details are omitted here forsimplicity.

Because an apparatus embodiment basically corresponds to a methodembodiment, for related parts, references can be made to relateddescriptions in the method embodiment. The previously describedapparatus embodiment is merely an example. The modules described asseparate parts may or may not be physically separate, and partsdisplayed as modules may or may not be physical modules, may be locatedin one position, or may be distributed on a number of network modules.Some or all of the modules can be selected based on actual demands toachieve the objectives of the solutions of the specification. A personof ordinary skill in the art can understand and implement theembodiments of the present application without creative efforts.

Referring again to FIG. 16B, it can be interpreted as illustrating aninternal functional module and a structure of a data processingapparatus for allocating M resources subject to L constraints. Anexecution body in essence can be an electronic device, and theelectronic device includes the following: one or more processors and amemory configured to store an executable instruction of the one or moreprocessors.

The one or more processors are configured to receive data representing Lconstraints, wherein each of the L constraints corresponds to a subsetof M resources and restricts a respective maximum number C of resourcesto be selected among the subset of the M resources, wherein any subsetcorresponding to a first constraint out of the L constraints has nocommon resource with any other subset corresponding to a secondconstraint out of the L constraints, unless the subset corresponding tothe first constraint is a subset of the other subset corresponding tothe second constraint; determine a topological ordering of the Lconstraints, wherein the topological ordering defines a traverse orderof the L constraints; select all the M resources as an initialselection; remove resources from the initial selection by traversingeach constraint in the topological ordering of the L constraints,wherein the remove includes: for a constraint l of the L constraintscorresponding to a subset S_(l) of the M resources and restricts amaximum number C_(l) of resources to be selected out of the subsetS_(l), determine currently selected resources in the subset S_(l); andunselect, among the currently selected resources in the subset S_(l),any resource that is not one of C_(l) resources that have top C_(l)rewards among the currently selected resources in the subset S_(l); andallocate the selected resources after traverse all the L constraints inthe topological ordering of the L constraints.

Optionally, the data representing the L constraints includes L indexsubsets, each of the L index subsets corresponding to a respective oneof the L constraints and including indexes of a subset of the Mresources corresponding to the respective one of the L constraints; andthe subset S_(l) includes indexes of a subset of the M resourcescorresponding to the constraint l.

Optionally, the data representing the L constraints includes datarepresenting each of the L constraints by a node in a directed acyclicgraph (DAG) data structure, and an edge of the DAG represents arelationship between two of the L constraints corresponding to two nodesof the edge.

Optionally, the one or more processors are configured to obtain datarepresenting M rewards, each reward corresponding to one of the Mresources respectively.

Optionally, the one or more processors are configured to sort the Mresources in a non-increase order according to the M rewardscorresponding to the M resources before traversing each subset in the Lsubsets in the topological ordering of the L subsets; and whereinunselecting, among the currently selected resources in the subset S_(l),any resource that is not one of C_(l) resources that have top C_(l)rewards among the currently selected resources in the subset S_(l)includes unselecting the any resource that is not one of C_(l) resourcesthat have top C_(l) rewards among the currently selected resources inthe subset S_(l) according to the non-increase order of the M resources.

Optionally, the M rewards are M cost-adjusted rewards respectively,wherein each of the M cost-adjusted rewards includes a reward minus acost if one of the M resources is selected.

Optionally, some or all of the above operations are performed for one ofN users in performing resource allocation of the M resources among the Nusers into K pools subject to K global constraints and the Lconstraints, wherein each of the L constraints restricts a respectivemaximum number of resources out of the M resources to be selected by asingle user, and each of the K global constraints limits resourcesallocated for each pool across the N users Optionally, L is in an orderof billions or larger.

FIG. 16C is a diagram of on example of modules of an apparatus 1600 c inaccordance with embodiments of this specification. The apparatus 1600 ccan be an example embodiment of a data processing apparatus forperforming resource allocation of M resources among N users into K poolsby solving a knapsack problem (KP) using a distributed computing systemthat includes a number of individual computer-implemented solvers, theKP subject to K global constraints and L local constraints. Theapparatus 1600 c can correspond to the embodiments described above, andthe apparatus 1600 c includes the following: a receiving module 1601 cfor receiving data representing the K global constraints and the L localconstraints, wherein each of the K global constraints restricts arespective maximum per-pool cost of the M resources across two or moreusers, and each of the L local constraints restricts a per-userselection of the M resources; a transforming module 1602 c fortransforming the KP into a dual problem using K dual multipliers, eachof the K dual multipliers corresponding to a respective one of the Kglobal constraints; a decomposing module 1603 c for decomposing the dualproblem into N sub-problems, each of the N sub-problems corresponding toa respective one of the N users and subject to the L local constraintswith respect to (w.r.t.) the respective one of the N users; a solvingmodule 1604 c for performing two or more iterations in solving the dualproblem, a determining module 1605 c for, in one iteration, for eachdual multiplier corresponding to a global constraint corresponding to apool that restricts a maximum per-pool cost of the M resources acrosstwo or more users, determining an updated dual multiplier for the globalconstraint corresponding to the pool to be a non-negative threshold suchthat: a total cost of resources selected across the N users for the poolcorresponding to a dual multiplier that is less than the non-negativethreshold exceeds the maximum per-pool cost; and a total cost ofresources selected across the N users for the pool corresponding to adual multiplier that is not less than the non-negative threshold doesnot exceed the maximum per-pool cost; and a computing module 1606 c forcomputing M decision variables of each of the N users corresponding tothe updated dual multiplier in solving each of the N sub-problemscorresponding to the each of the N users, wherein each of the M decisionvariables indicates whether or not to select a respective one of the Mresources by the each of the N users.

In an optional embodiment, N is in an order of billions or larger, K issmaller than L, L is in an order of billions or larger.

In an optional embodiment, each of the L constraints corresponds to asubset of M resources and restricts a respective maximum number C ofresources to be selected among the subset of the M resources, whereinany subset corresponding to a first constraint out of the L constraintshas no common resource with any other subset corresponding to a secondconstraint out of the L constraints, unless the subset corresponding tothe first constraint is a subset of the other subset corresponding tothe second constraint.

In an optional embodiment, the apparatus 1600 c further includes:another computing module for, in response to determining that aconvergence condition is met, computing M decision variables of eachuser of the N users corresponding to the K dual multipliers; and anallocating module allocating the M resources among the N users accordingto the M decision variables of each user of the N users.

In an optional embodiment, the apparatus 1600 c further includes:another determining module for determining initial values of the K dualmultipliers, wherein the initial values of the K dual multipliers arecomputed based on a sampled version of the KP for resource allocationamong S users into the K pools subject to K sampled global constraintsand the L local constraints, wherein the S users are sampled from the Nusers, and each of the K sampled global constraints restricts arespective scaled-down maximum total cost of resources across the Susers for each of the K pools.

In an optional embodiment, solving each of the N sub-problemscorresponding to the each of the N users includes: for a sub-problemcorresponding to a user and for each dual multiplier corresponding tothe global constraint corresponding to the pool, computing candidates ofthe dual multiplier; sorting the candidates of the dual multiplier in anon-increasing order; traversing the candidates of the dual multiplierin the non-increasing order: for each candidate of the dual multiplier;computing M decision variables of the user corresponding to thecandidate of the dual multiplier; and computing an incremental cost ofselected resources based on the M decision variables of the usercorresponding to the candidate of the dual multiplier relative to a costof selected resources based on the M decision variables of the usercorresponding to a candidate of the dual multiplier preceding thecandidate of the dual multiplier in the non-increasing order; andwherein a total cost of resources selected across the N users for thepool corresponding to a certain dual multiplier is computed by summingincremental costs of selected resources of the user corresponding to allcandidates of the dual multiplier that are larger than or equal to thecertain dual multiplier.

In an optional embodiment, computing candidates of the dual multiplierincludes computing pair-wise intersection points of M linear functionsof the dual multiplier, wherein the M linear functions of the dualmultiplier represent M cost-adjusted rewards of the M resources giventhe dual multiplier.

In an optional embodiment, the apparatus 1600 c further includes:another computing module for computing the non-negative threshold basedon the candidates of the dual multiplier sorted in the non-increasingorder and a respective total cost of resources selected across the Nusers for the pool corresponding to one of the candidates of the dualmultiplier in the non-increasing order.

In an optional embodiment, the apparatus 1600 c further includes: aninterpolating module for performing interpolation on the candidates ofthe dual multiplier sorted in the non-increasing order for the computingthe non-negative threshold.

In an optional embodiment, the apparatus 1600 c further includes: agrouping module for grouping the candidates of the dual multiplier intoa plurality of non-uniform budgets; another computing module forcomputing sums of incremental costs of selected resources of the usercorresponding to respective candidates of the dual multiplier in theplurality of non-uniform budgets; and another computing module forcomputing the non-negative threshold based on the sums of incrementalcosts.

The system, apparatus, module, or unit illustrated in the previousembodiments can be implemented by using a computer chip or an entity, orcan be implemented by using a product having a certain function. Atypical embodiment device is a computer, and the computer can be apersonal computer, a laptop computer, a cellular phone, a camera phone,a smartphone, a personal digital assistant, a media player, a navigationdevice, an email receiving and sending device, a game console, a tabletcomputer, a wearable device, or any combination of these devices.

For an embodiment process of functions and roles of each module in theapparatus, references can be made to an embodiment process ofcorresponding steps in the previous method. Details are omitted here forsimplicity.

Because an apparatus embodiment basically corresponds to a methodembodiment, for related parts, references can be made to relateddescriptions in the method embodiment. The previously describedapparatus embodiment is merely an example. The modules described asseparate parts may or may not be physically separate, and partsdisplayed as modules may or may not be physical modules, may be locatedin one position, or may be distributed on a number of network modules.Some or all of the modules can be selected based on actual demands toachieve the objectives of the solutions of the specification. A personof ordinary skill in the art can understand and implement theembodiments of the present application without creative efforts.

Referring again to FIG. 16C, it can be interpreted as illustrating aninternal functional module and a structure of a data processingapparatus, such as a distributed computing system that includes a numberof individual computer-implemented solvers, for performing resourceallocation of M resources among N users into K pools by solving aknapsack problem (KP) subject to K global constraints and L localconstraints. An execution body in essence can be an electronic device,and the electronic device includes the following: one or more processorsand a memory configured to store an executable instruction of the one ormore processors.

The one or more processors are configured to receive data representingthe K global constraints and the L local constraints, wherein each ofthe K global constraints restricts a respective maximum per-pool cost ofthe M resources across two or more users, and each of the L localconstraints restricts a per-user selection of the M resources;transforms the KP into a dual problem using K dual multipliers, each ofthe K dual multipliers corresponding to a respective one of the K globalconstraints; decomposes the dual problem into N sub-problems, each ofthe N sub-problems corresponding to a respective one of the N users andsubject to the L local constraints with respect to (w.r.t.) therespective one of the N users; performs two or more iterations insolving the dual problem, wherein in one iteration, for each dualmultiplier corresponding to a global constraint corresponding to a poolthat restricts a maximum per-pool cost of the M resources across two ormore users: determines an updated dual multiplier for the globalconstraint corresponding to the pool to be a non-negative threshold suchthat: a total cost of resources selected across the N users for the poolcorresponding to a dual multiplier that is less than the non-negativethreshold exceeds the maximum per-pool cost; and a total cost ofresources selected across the N users for the pool corresponding to adual multiplier that is not less than the non-negative threshold doesnot exceed the maximum per-pool cost; and compute M decision variablesof each of the N users corresponding to the updated dual multiplier insolving each of the N sub-problems corresponding to the each of the Nusers, wherein each of the M decision variables indicates whether or notto select a respective one of the M resources by the each of the Nusers.

Optionally, N is in an order of billions or larger, K is smaller than L,L is in an order of billions or larger.

Optionally, each of the L constraints corresponds to a subset of Mresources and restricts a respective maximum number C of resources to beselected among the subset of the M resources, wherein any subsetcorresponding to a first constraint out of the L constraints has nocommon resource with any other subset corresponding to a secondconstraint out of the L constraints, unless the subset corresponding tothe first constraint is a subset of the other subset corresponding tothe second constraint.

Optionally, the one or more processors are configured to, in response todetermining that a convergence condition is met, compute M decisionvariables of each user of the N users corresponding to the K dualmultipliers; and allocate the M resources among the N users according tothe M decision variables of each user of the N users.

Optionally, the one or more processors are configured to determineinitial values of the K dual multipliers, wherein the initial values ofthe K dual multipliers are computed based on a sampled version of the KPfor resource allocation among S users into the K pools subject to Ksampled global constraints and the L local constraints, wherein the Susers are sampled from the N users, and each of the K sampled globalconstraints restricts a respective scaled-down maximum total cost ofresources across the S users for each of the K pools.

Optionally, solving each of the N sub-problems corresponding to the eachof the N users includes: for a sub-problem corresponding to a user andfor each dual multiplier corresponding to the global constraintcorresponding to the pool, computing candidates of the dual multiplier;sorting the candidates of the dual multiplier in a non-increasing order;traversing the candidates of the dual multiplier in the non-increasingorder: for each candidate of the dual multiplier, computing M decisionvariables of the user corresponding to the candidate of the dualmultiplier; and computing an incremental cost of selected resourcesbased on the M decision variables of the user corresponding to thecandidate of the dual multiplier relative to a cost of selectedresources based on the M decision variables of the user corresponding toa candidate of the dual multiplier preceding the candidate of the dualmultiplier in the non-increasing order; and wherein a total cost ofresources selected across the N users for the pool corresponding to acertain dual multiplier is computed by summing incremental costs ofselected resources of the user corresponding to all candidates of thedual multiplier that are larger than or equal to the certain dualmultiplier.

Optionally, computing candidates of the dual multiplier includescomputing pair-wise intersections points of M linear functions of thedual multiplier, wherein the M linear functions of the dual multiplierrepresent M cost-adjusted rewards of the M resources given the dualmultiplier.

Optionally, the one or more processors are configured to compute thenon-negative threshold based on the candidates of the dual multipliersorted in the non-increasing order and a respective total cost ofresources selected across the N users for the pool corresponding to oneof the candidates of the dual multiplier in the non-increasing order.

Optionally, the one or more processors are configured to performinterpolation on the candidates of the dual multiplier sorted in thenon-increasing order for the computing the non-negative threshold.

Optionally, the one or more processors are configured to group thecandidates of the dual multiplier into a plurality of non-uniformbudgets; compute sums of incremental costs of selected resources of theuser corresponding to respective candidates of the dual multiplier inthe plurality of non-uniform budgets; and compute the non-negativethreshold based on the sums of incremental costs.

The techniques described in this specification produce one or moretechnical effects. In some embodiments, a distributed system that iscapable of solving billion-scale real-world knapsack problems (e.g.,with billions of decision variables and constraints) is described. Thedistributed system is developed based on a slightly generalizedformulation of KPs and hence can be applied to solve other variants ofKPs. By introducing local constraints and treating them separately fromglobal ones, flexible business requirements on each user of resourcescan be enforced, without increasing the complexity of satisfying globalconstraints.

As an example, the described techniques can solve an optimizationproblem for allocating limited resources to each user efficiently,without sacrificing business metrics such as user satisfaction. Oneexample of a use case is marketing campaigns that target hundreds ofmillions of users. Each campaign may have a global resource limit, aswell as, a separate resource limit for each promotion channel. Dependingon the nature of the campaign operations, the resources referred herecan be in flexible forms, such as monetary and non-monetary promotions.One example of a non-monetary resource constraint is that the totalnumber of mobile App page views allocated to a marketing event may belimited. In some embodiments, individual users may have per-user localconstraints. For example, a targeted user cannot participate in morethan two promotion channels. In some embodiments, both the decisionvariables and local constraints can reach billion level while the numberof global constraints is often around hundreds.

This specification also describes a greedy algorithm andcomputer-implemented solver for solving an IP problem subject toconstraints having a hierarchal structure. The described greedyalgorithm and computer-implemented solver can leverage the hierarchalstructure of the constraints and solve the IP problem more efficiently,for example, with a polynomial time complexity. The greedy algorithm andcomputer-implemented solver can be used to find solutions to real-worldscenarios such as resource allocation, product/service promotion andmarketing that can be modeled or represented by an IP problem in a moreefficient manner.

This specification further describes a synchronous coordinate descent(SCD) algorithm for updating dual multipliers. Unlike existing updatingalgorithms that requires a hyper-parameter a that needs to be choseneither manually or programmatically, which can be practically cumbersomeor computationally intensive, especially for large-scale KPs, the SCDalgorithm can be implemented without such a hyper-parameter, thusimproving computational efficiency and reducing the computational load.In some embodiments, the SCD algorithm is less prone to constraintviolation, thus provide better solutions to the underlying optimizationproblems.

In some embodiments, the SCD algorithm can be used in the distributedsystem using the MapReduce or other distributed computing framework ormodel. The distributed SCD algorithm can be implemented using commondistributed platforms such as MPI, HADOOP, and SPARK.

In some embodiments, the described techniques can be generalized tooptimize large-scale non-linear problems. The objective function ascurrently formulated in is linear in the decision variables Thedistributed solution framework discussed can be extended to optimize anon-linear objective function, as long as it is decomposable withrespect to the decision variables (or users of decision variables).

Described embodiments of the subject matter can include one or morefeatures, alone or in combination. For example, in a first embodiment, acomputer-implemented method for performing resource allocation of Mresources among N users into K pools by solving a knapsack problem (KP)using a distributed computing system that includes a number ofindividual computer-implemented solvers, the KP subject to K globalconstraints and L local constraints. The method includes: receiving datarepresenting the K global constraints and the L local constraints,wherein K is smaller than L, each of the K global constraints restrictsa respective maximum per-pool cost of the M resources across two or moreusers, and each of the L local constraints restricts a per-userselection of the M resources; decomposing the KP into N sub-problemsusing K dual multipliers, each of the N sub-problems corresponding to arespective one of the N users and subject to the L local constraintswith respect to (w.r.t.) the corresponding user, wherein N is in anorder of billions or larger, wherein each of the K dual multiplierscorresponds to a respective one of the K global constraints; determiningthe number of individual computer-implemented solvers for solving the Nsub-problems; distributing the N sub-problems among the number ofindividual computer-implemented solvers by assigning each sub-problem toa respective computer-implemented solver; and solving the KP by thedistributed computing system by performing two or more iterations, and,in one iteration, the method including: solving each of the Nsub-problems by the computer-implemented solver to which the sub-problemwas assigned independently, wherein solving each of the N sub-problemsincludes computing an amount of each of the M resources to be allocatedto the corresponding user of the N user; and updating the K dualmultipliers based on the amount of each of the M resources to beallocated to the corresponding user of the N user.

The foregoing and other described embodiments can each, optionally,include one or more of the following features:

A first feature, combinable with any of the following features, whereinL is in an order of billions or larger.

A second feature, combinable with any of the following features,wherein: determining the number of individual computer-implementedsolvers for solving the N sub-problems includes determining the numberof individual computer-implemented solvers for solving the Nsub-problems based on a specified computation time for solving the KP;and solving each of the N sub-problems by the computer-implementedsolver to which the sub-problem was assigned independently includessolving two or more of the N sub-problems by respectivecomputer-implemented solvers to which the two or more of the Nsub-problems were assigned in parallel.

A third feature, combinable with any of the following features, wherein:determining the number of individual computer-implemented solvers forsolving the N sub-problems includes determining the number of individualcomputer-implemented solvers to be N, and solving each of the Nsub-problems by one of the number of individual computer-implementedsolvers independently includes solving each of the N sub-problems by arespective one of the N individual computer-implemented solvers inparallel.

A fourth feature, combinable with any of the following features, whereinsolving each of the N sub-problems by the computer-implemented solver towhich the sub-problem was assigned independently includes solving aninteger programming (IP) problem by an IP computer-implemented solverindependently.

A fifth feature, combinable with any of the following features, furtherincluding: determining whether a convergence condition is met based onthe K dual multipliers; in response to determining that a convergencecondition is met based on the K dual multipliers, solving each of the Nsub-problems based on the K dual multipliers by one of the number ofindividual computer-implemented solvers independently, wherein solvingeach of the N sub-problems includes computing an amount of each of the Mresources to be allocated to the corresponding user of the N user; andallocating the amount of each of the M resources to the correspondinguser of the N user.

A sixth feature, combinable with any of the following features, whereinsolving each of the N sub-problems further includes computing, for eachof the K pools, a per-pool cost of the M resources by the correspondinguser based on the amount of each of the M resources to be allocated tothe corresponding user.

A seventh feature, combinable with any of the following features,wherein, in one iteration, the method further includes: computing, foreach of the K pools, a per-pool cost of the M resources across the Nusers based on the amount of each of the M resources to be allocated tothe corresponding user of the N user; and wherein updating the K dualmultipliers based on the amount of each of the M resources to beallocated to the corresponding user of the N user includes updating eachof the K dual multipliers w.r.t. a corresponding pool based on adifference between a maximum per-pool cost of the M resources across twoor more users for the corresponding pool restricted by a correspondingglobal constraint and a per-pool cost of the M resources across the Nusers for the corresponding pool computed based on the amount of each ofthe M resources to be allocated to the corresponding user of the N user.

For example, in a second embodiment, a computer-implemented method forallocating M resources subject to L constraints, the method including:receiving data representing L constraints, wherein each of the Lconstraints corresponds to a subset of M resources and restricts arespective maximum number C of resources to be selected among the subsetof the M resources, wherein any subset corresponding to a firstconstraint out of the L constraints has no common resource with anyother subset corresponding to a second constraint out of the Lconstraints, unless the subset corresponding to the first constraint isa subset of the other subset corresponding to the second constraint;determining a topological ordering of the L constraints, wherein thetopological ordering defines a traversing order of the L constraints;selecting all the M resources as an initial selection; removingresources from the initial selection by traversing each constraint inthe topological ordering of the L constraints, wherein the removingincludes: for a constraint l of the L constraints corresponding to asubset S_(l) of the M resources and restricting a maximum number C_(l)of resources to be selected out of the subset S_(l), determiningcurrently selected resources in the subset S_(l); and unselecting, amongthe currently selected resources in the subset S_(l), any resource thatis not one of C_(l) resources that have top C_(l) rewards among thecurrently selected resources in the subset S_(l); and allocating theselected resources after traversing all the L constraints in thetopological ordering of the L constraints.

The foregoing and other described embodiments can each, optionally,include one or more of the following features:

A first feature, combinable with any of the following features, wherein:the data representing the L constraints includes L index subsets, eachof the L index subsets corresponds to a respective one of the Lconstraints and includes indexes of a subset of the M resourcescorresponding to the respective one of the L constraints; and the subsetS_(l) includes indexes of a subset of the M resources corresponding tothe constraint l.

A second feature, combinable with any of the following features, whereinthe data representing the L constraints includes data representing eachof the L constraints by a node in a directed acyclic graph (DAG) datastructure, and an edge of the DAG represents a relationship between twoof the L constraints corresponding to two nodes of the edge.

A third feature, combinable with any of the following features, furtherincluding: obtaining data representing M rewards, each rewardcorresponding to one of the M resources respectively.

A fourth feature, combinable with any of the following features, furtherincluding: sorting the M resources in a non-increasing order accordingto the M rewards corresponding to the M resources before traversing eachsubset in the L subsets in the topological ordering of the L subsets;and wherein unselecting, among the currently selected resources in thesubset S_(l), any resource that is not one of C_(l) resources that havetop C_(l) rewards among the currently selected resources in the subsetS_(l) includes unselecting the any resource that is not one of C_(l)resources that have top C_(l) rewards among the currently selectedresources in the subset S_(l) according to the non-increasing order ofthe M resources.

A fifth feature, combinable with any of the following features, whereinthe M rewards are M cost-adjusted rewards respectively, wherein each ofthe M cost-adjusted rewards includes a reward minus a cost if one of theM resources is selected.

A sixth feature, combinable with any of the following features, whereinthe method is performed for one of N users in performing resourceallocation of the M resources among the N users into K pools subject toK global constraints and the L constraints, wherein each of the Lconstraints restricts a respective maximum number of resources out ofthe M resources to be selected by a single user, and each of the Kglobal constraints limits resources allocated for each pool across the Nusers.

For example, in a third embodiment, a computer-implemented method forperforming resource allocation, including: using a distributed computingsystem that includes a number of individual computer-implemented solversfor performing resource allocation of M resources among N users into Kpools by solving a knapsack problem (KP) subject to K global constraintsand L local constraints: receiving data representing the K globalconstraints and the L local constraints, wherein each of the K globalconstraints restricts a respective maximum per-pool cost of the Mresources across two or more users, and each of the L local constraintsrestricts a per-user selection of the M resources; transforming the KPinto a dual problem using K dual multipliers, each of the K dualmultipliers corresponding to a respective one of the K globalconstraints; decomposing the dual problem into N sub-problems, each ofthe N sub-problems corresponding to a respective one of the N users andsubject to the L local constraints with respect to (w.r.t.) therespective one of the N users; performing two or more iterations insolving the dual problem, wherein in one iteration, for each dualmultiplier corresponding to a global constraint corresponding to a poolthat restricts a maximum per-pool cost of the M resources across two ormore users: determining an updated dual multiplier for the globalconstraint corresponding to the pool to be a non-negative threshold suchthat: a total cost of resources selected across the N users for the poolcorresponding to a dual multiplier that is less than the non-negativethreshold exceeds the maximum per-pool cost; and a total cost ofresources selected across the N users for the pool corresponding to adual multiplier that is not less than the non-negative threshold doesnot exceed the maximum per-pool cost; and computing M decision variablesof each of the N users corresponding to the updated dual multiplier insolving each of the N sub-problems corresponding to the each of the Nusers, wherein each of the M decision variables indicates whether or notto select a respective one of the M resources by the each of the Nusers.

The foregoing and other described embodiments can each, optionally,include one or more of the following features:

A first feature, combinable with any of the following features, whereinN is in an order of billions or larger, K is smaller than L, L is in anorder of billions or larger.

A second feature, combinable with any of the following features, whereineach of the L constraints corresponds to a subset of M resources andrestricts a respective maximum number C of resources to be selectedamong the subset of the M resources, wherein any subset corresponding toa first constraint out of the L constraints has no common resource withany other subset corresponding to a second constraint out of the Lconstraints, unless the subset corresponding to the first constraint isa subset of the other subset corresponding to the second constraint.

A third feature, combinable with any of the following features, furtherincluding: in response to determining that a convergence condition ismet, computing M decision variables of each user of the N userscorresponding to the K dual multipliers; and allocating the M resourcesamong the N users according to the M decision variables of each user ofthe N users.

A fourth feature, combinable with any of the following features, furtherincluding determining initial values of the K dual multipliers, whereinthe initial values of the K dual multipliers are computed based on asampled version of the KP for resource allocation among S users into theK pools subject to K sampled global constraints and the L localconstraints, wherein the S users are sampled from the N users, and eachof the K sampled global constraints restricts a respective scaled-downmaximum total cost of resources across the S users for each of the Kpools.

A fifth feature, combinable with any of the following features, whereinsolving each of the N sub-problems corresponding to the each of the Nusers includes: for a sub-problem corresponding to a user and for eachdual multiplier corresponding to the global constraint corresponding tothe pool, computing candidates of the dual multiplier; sorting thecandidates of the dual multiplier in a non-increasing order; traversingthe candidates of the dual multiplier in the non-increasing order: foreach candidate of the dual multiplier, computing M decision variables ofthe user corresponding to the candidate of the dual multiplier; andcomputing an incremental cost of selected resources based on the Mdecision variables of the user corresponding to the candidate of thedual multiplier relative to a cost of selected resources based on the Mdecision variables of the user corresponding to a candidate of the dualmultiplier preceding the candidate of the dual multiplier in thenon-increasing order; and wherein a total cost of resources selectedacross the N users for the pool corresponding to a certain dualmultiplier is computed by summing incremental costs of selectedresources of the user corresponding to all candidates of the dualmultiplier that are larger than or equal to the certain dual multiplier.

A sixth feature, combinable with any of the following features, whereincomputing candidates of the dual multiplier includes computing pair-wiseintersections points of M linear functions of the dual multiplier,wherein the M linear functions of the dual multiplier represent Mcost-adjusted rewards of the M resources given the dual multiplier.

A seventh feature, combinable with any of the following features,further including computing the non-negative threshold based on thecandidates of the dual multiplier sorted in the non-increasing order anda respective total cost of resources selected across the N users for thepool corresponding to one of the candidates of the dual multiplier inthe non-increasing order.

An eighth feature, combinable with any of the following features,further including performing interpolation on the candidates of the dualmultiplier sorted in the non-increasing order for the computing thenon-negative threshold.

A ninth feature, combinable with any of the following features, furtherincluding: grouping the candidates of the dual multiplier into aplurality of non-uniform budgets; computing sums of incremental costs ofselected resources of the user corresponding to respective candidates ofthe dual multiplier in the plurality of non-uniform budgets; andcomputing the non-negative threshold based on the sums of incrementalcosts.

Embodiments of the subject matter and the actions and operationsdescribed in this specification can be implemented in digital electroniccircuitry, in tangibly-embodied computer software or firmware, incomputer hardware, including the structures disclosed in thisspecification and their structural equivalents, or in combinations ofone or more of them. Embodiments of the subject matter described in thisspecification can be implemented as one or more computer programs, e.g.,one or more modules of computer program instructions, encoded on acomputer program carrier, for execution by, or to control the operationof, data processing apparatus. For example, a computer program carriercan include one or more computer-readable storage media that haveinstructions encoded or stored thereon. The carrier may be a tangiblenon-transitory computer-readable medium, such as a magnetic, magnetooptical, or optical disk, a solid state drive, a random access memory(RAM), a read-only memory (ROM), or other types of media. Alternatively,or in addition, the carrier may be an artificially generated propagatedsignal, e.g., a machine-generated electrical, optical, orelectromagnetic signal that is generated to encode information fortransmission to suitable receiver apparatus for execution by a dataprocessing apparatus. The computer storage medium can be or be part of amachine-readable storage device, a machine-readable storage substrate, arandom or serial access memory device, or a combination of one or moreof them. A computer storage medium is not a propagated signal.

A computer program, which may also be referred to or described as aprogram, software, a software application, an app, a module, a softwaremodule, an engine, a script, or code, can be written in any form ofprogramming language, including compiled or interpreted languages, ordeclarative or procedural languages; and it can be deployed in any form,including as a stand-alone program or as a module, component, engine,subroutine, or other unit suitable for executing in a computingenvironment, which environment may include one or more computersinterconnected by a data communication network in one or more locations.

A computer program may, but need not, correspond to a file in a filesystem. A computer program can be stored in a portion of a file thatholds other programs or data, e.g., one or more scripts stored in amarkup language document, in a single file dedicated to the program inquestion, or in multiple coordinated files, e.g., files that store oneor more modules, sub programs, or portions of code.

Processors for execution of a computer program include, by way ofexample, both general- and special-purpose microprocessors, and any oneor more processors of any kind of digital computer. Generally, aprocessor will receive the instructions of the computer program forexecution as well as data from a non-transitory computer-readable mediumcoupled to the processor.

The term “data processing apparatus” encompasses all kinds ofapparatuses, devices, and machines for processing data, including by wayof example a programmable processor, a computer, or multiple processorsor computers. Data processing apparatus can include special-purposelogic circuitry, e.g., an FPGA (field programmable gate array), an ASIC(application specific integrated circuit), or a GPU (graphics processingunit). The apparatus can also include, in addition to hardware, codethat creates an execution environment for computer programs, e.g., codethat constitutes processor firmware, a protocol stack, a databasemanagement system, an operating system, or a combination of one or moreof them.

The processes and logic flows described in this specification can beperformed by one or more computers or processors executing one or morecomputer programs to perform operations by operating on input data andgenerating output. The processes and logic flows can also be performedby special-purpose logic circuitry, e.g., an FPGA, an ASIC, or a GPU, orby a combination of special-purpose logic circuitry and one or moreprogrammed computers.

Computers suitable for the execution of a computer program can be basedon general or special-purpose microprocessors or both, or any other kindof central processing unit. Generally, a central processing unit willreceive instructions and data from a read only memory or a random accessmemory or both. Elements of a computer can include a central processingunit for executing instructions and one or more memory devices forstoring instructions and data. The central processing unit and thememory can be supplemented by, or incorporated in, special-purpose logiccircuitry.

Generally, a computer will also include, or be operatively coupled toreceive data from or transfer data to one or more storage devices. Thestorage devices can be, for example, magnetic, magneto optical, oroptical disks, solid state drives, or any other type of non-transitory,computer-readable media. However, a computer need not have such devices.Thus, a computer may be coupled to one or more storage devices, such as,one or more memories, that are local and/or remote. For example, acomputer can include one or more local memories that are integralcomponents of the computer, or the computer can be coupled to one ormore remote memories that are in a cloud network. Moreover, a computercan be embedded in another device, e.g., a mobile telephone, a personaldigital assistant (PDA), a mobile audio or video player, a game console,a Global Positioning System (GPS) receiver, or a portable storagedevice, e.g., a universal serial bus (USB) flash drive, to name just afew.

Components can be “coupled to” each other by being commutatively such aselectrically or optically connected to one another, either directly orvia one or more intermediate components. Components can also be “coupledto” each other if one of the components is integrated into the other.For example, a storage component that is integrated into a processor(e.g., an L2 cache component) is “coupled to” the processor.

To provide for interaction with a user, embodiments of the subjectmatter described in this specification can be implemented on, orconfigured to communicate with, a computer having a display device,e.g., a LCD (liquid crystal display) monitor, for displaying informationto the user, and an input device by which the user can provide input tothe computer, e.g., a keyboard and a pointing device, e.g., a mouse, atrackball or touchpad. Other kinds of devices can be used to provide forinteraction with a user as well; for example, feedback provided to theuser can be any form of sensory feedback, e.g., visual feedback,auditory feedback, or tactile feedback; and input from the user can bereceived in any form, including acoustic, speech, or tactile input. Inaddition, a computer can interact with a user by sending documents toand receiving documents from a device that is used by the user; forexample, by sending web pages to a web browser on a user's device inresponse to requests received from the web browser, or by interactingwith an app running on a user device, e.g., a smartphone or electronictablet. Also, a computer can interact with a user by sending textmessages or other forms of message to a personal device, e.g., asmartphone that is running a messaging application, and receivingresponsive messages from the user in return.

This specification uses the term “configured to” in connection withsystems, apparatus, and computer program components. For a system of oneor more computers to be configured to perform particular operations oractions means that the system has installed on it software, firmware,hardware, or a combination of them that in operation cause the system toperform the operations or actions. For one or more computer programs tobe configured to perform particular operations or actions means that theone or more programs include instructions that, when executed by dataprocessing apparatus, cause the apparatus to perform the operations oractions. For special-purpose logic circuitry to be configured to performparticular operations or actions means that the circuitry has electroniclogic that performs the operations or actions.

While this specification contains many specific embodiment details,these should not be construed as limitations on the scope of what isbeing claimed, which is defined by the claims themselves, but rather asdescriptions of features that may be specific to particular embodiments.Certain features that are described in this specification in the contextof separate embodiments can also be realized in combination in a singleembodiment. Conversely, various features that are described in thecontext of a single embodiments can also be realized in multipleembodiments separately or in any suitable subcombination. Moreover,although features may be described above as acting in certaincombinations and even initially be claimed as such, one or more featuresfrom a claimed combination can in some cases be excised from thecombination, and the claim may be directed to a subcombination orvariation of a subcombination.

Similarly, while operations are depicted in the drawings and recited inthe claims in a particular order, this should not be understood asrequiring that such operations be performed in the particular ordershown or in sequential order, or that all illustrated operations beperformed, to achieve desirable results. In certain circumstances,multitasking and parallel processing may be advantageous. Moreover, theseparation of various system modules and components in the embodimentsdescribed above should not be understood as requiring such separation inall embodiments, and it should be understood that the described programcomponents and systems can generally be integrated together in a singlesoftware product or packaged into multiple software products.

Particular embodiments of the subject matter have been described. Otherembodiments are within the scope of the following claims. For example,the actions recited in the claims can be performed in a different orderand still achieve desirable results. As one example, the processesdepicted in the accompanying figures do not necessarily require theparticular order shown, or sequential order, to achieve desirableresults. In some cases, multitasking and parallel processing may beadvantageous.

What is claimed is:
 1. A computer-implemented method for performingresource allocation, comprising: using a distributed computing systemthat includes a number of individual computer-implemented solvers forperforming resource allocation of M resources among N users into K poolsby solving a knapsack problem (KP) subject to K global constraints and Llocal constraints: receiving data representing the K global constraintsand the L local constraints, wherein each of the K global constraintsrestricts a respective maximum per-pool cost of the M resources acrosstwo or more users, and each of the L local constraints restricts aper-user selection of the M resources; transforming the KP into a dualproblem using K dual multipliers, each of the K dual multiplierscorresponding to a respective one of the K global constraints;decomposing the dual problem into N sub-problems, each of the Nsub-problems corresponding to a respective one of the N users andsubject to the L local constraints with respect to (w.r.t.) therespective one of the N users; performing two or more iterations insolving the dual problem, wherein in one iteration, for each dualmultiplier corresponding to a global constraint corresponding to a poolthat restricts a maximum per-pool cost of the M resources across two ormore users: determining an updated dual multiplier for the globalconstraint corresponding to the pool to be a non-negative threshold suchthat: a total cost of resources selected across the N users for the poolcorresponding to a dual multiplier that is less than the non-negativethreshold exceeds the maximum per-pool cost; and a total cost ofresources selected across the N users for the pool corresponding to adual multiplier that is not less than the non-negative threshold doesnot exceed the maximum per-pool cost; and computing M decision variablesof each of the N users corresponding to the updated dual multiplier insolving each of the N sub-problems corresponding to the each of the Nusers by one of the number of individual computer-implemented solvers,wherein each of the M decision variables indicates whether or not toselect a respective one of the M resources by the each of the N users.2. The method of claim 1, wherein N is in an order of billions orlarger, K is smaller than L, L is in an order of billions or larger. 3.The method of claim 1, wherein each of the L constraints corresponds toa subset of M resources and restricts a respective maximum number C ofresources to be selected among the subset of the M resources, whereinany subset corresponding to a first constraint out of the L constraintshas no common resource with any other subset corresponding to a secondconstraint out of the L constraints, unless the subset corresponding tothe first constraint is a subset of the other subset corresponding tothe second constraint.
 4. The method of claim 1, further comprising: inresponse to determining that a convergence condition is met, computing Mdecision variables of each user of the N users corresponding to the Kdual multipliers; and allocating the M resources among the N usersaccording to the M decision variables of each user of the N users. 5.The method of claim 1, further comprising determining initial values ofthe K dual multipliers, wherein the initial values of the K dualmultipliers are computed based on a sampled version of the KP forresource allocation among S users into the K pools subject to K sampledglobal constraints and the L local constraints, wherein the S users aresampled from the N users, and each of the K sampled global constraintsrestricts a respective scaled-down maximum total cost of resourcesacross the S users for each of the K pools.
 6. The method of claim 1,wherein solving each of the N sub-problems corresponding to the each ofthe N users comprises: for a sub-problem corresponding to a user and foreach dual multiplier corresponding to the global constraintcorresponding to the pool, computing candidates of the dual multiplier;sorting the candidates of the dual multiplier in a non-increasing order;traversing the candidates of the dual multiplier in the non-increasingorder: for each candidate of the dual multiplier, computing M decisionvariables of the user corresponding to the candidate of the dualmultiplier; and computing an incremental cost of selected resourcesbased on the M decision variables of the user corresponding to thecandidate of the dual multiplier relative to a cost of selectedresources based on the M decision variables of the user corresponding toa candidate of the dual multiplier preceding the candidate of the dualmultiplier in the non-increasing order; and wherein a total cost ofresources selected across the N users for the pool corresponding to acertain dual multiplier is computed by summing incremental costs ofselected resources of the user corresponding to all candidates of thedual multiplier that are larger than or equal to the certain dualmultiplier.
 7. The method of claim 6, wherein computing candidates ofthe dual multiplier comprises computing pair-wise intersections pointsof M linear functions of the dual multiplier, wherein the M linearfunctions of the dual multiplier represent M cost-adjusted rewards ofthe M resources given the dual multiplier.
 8. The method of claim 6,further comprising computing the non-negative threshold based on thecandidates of the dual multiplier sorted in the non-increasing order anda respective total cost of resources selected across the N users for thepool corresponding to one of the candidates of the dual multiplier inthe non-increasing order.
 9. The method of claim 8, further comprisingperforming interpolation on the candidates of the dual multiplier sortedin the non-increasing order for the computing the non-negativethreshold.
 10. The method of claim 6, further comprising: grouping thecandidates of the dual multiplier into a plurality of non-uniformbudgets; computing sums of incremental costs of selected resources ofthe user corresponding to respective candidates of the dual multiplierin the plurality of non-uniform budgets; and computing the non-negativethreshold based on the sums of incremental costs.
 11. A non-transitory,computer-readable storage medium storing one or more instructionsexecutable by a computer system to perform operations for performingresource allocation, the operations comprising: using a distributedcomputing system that includes a number of individualcomputer-implemented solvers for performing resource allocation of Mresources among N users into K pools by solving a knapsack problem (KP)subject to K global constraints and L local constraints: receiving datarepresenting the K global constraints and the L local constraints,wherein each of the K global constraints restricts a respective maximumper-pool cost of the M resources across two or more users, and each ofthe L local constraints restricts a per-user selection of the Mresources; transforming the KP into a dual problem using K dualmultipliers, each of the K dual multipliers corresponding to arespective one of the K global constraints; decomposing the dual probleminto N sub-problems, each of the N sub-problems corresponding to arespective one of the N users and subject to the L local constraintswith respect to (w.r.t.) the respective one of the N users; performingtwo or more iterations in solving the dual problem, wherein in oneiteration, for each dual multiplier corresponding to a global constraintcorresponding to a pool that restricts a maximum per-pool cost of the Mresources across two or more users: determining an updated dualmultiplier for the global constraint corresponding to the pool to be anon-negative threshold such that: a total cost of resources selectedacross the N users for the pool corresponding to a dual multiplier thatis less than the non-negative threshold exceeds the maximum per-poolcost; and a total cost of resources selected across the N users for thepool corresponding to a dual multiplier that is not less than thenon-negative threshold does not exceed the maximum per-pool cost; andcomputing M decision variables of each of the N users corresponding tothe updated dual multiplier in solving each of the N sub-problemscorresponding to the each of the N users by one of the number ofindividual computer-implemented solvers, wherein each of the M decisionvariables indicates whether or not to select a respective one of the Mresources by the each of the N users.
 12. The non-transitory,computer-readable storage medium of claim 11, wherein N is in an orderof billions or larger, K is smaller than L, L is in an order of billionsor larger.
 13. The non-transitory, computer-readable storage medium ofclaim 11, wherein each of the L constraints corresponds to a subset of Mresources and restricts a respective maximum number C of resources to beselected among the subset of the M resources, wherein any subsetcorresponding to a first constraint out of the L constraints has nocommon resource with any other subset corresponding to a secondconstraint out of the L constraints, unless the subset corresponding tothe first constraint is a subset of the other subset corresponding tothe second constraint.
 14. The non-transitory, computer-readable storagemedium of claim 11, the operations further comprising: in response todetermining that a convergence condition is met, computing M decisionvariables of each user of the N users corresponding to the K dualmultipliers; and allocating the M resources among the N users accordingto the M decision variables of each user of the N users.
 15. Thenon-transitory, computer-readable storage medium of claim 11, theoperations further comprising determining initial values of the K dualmultipliers, wherein the initial values of the K dual multipliers arecomputed based on a sampled version of the KP for resource allocationamong S users into the K pools subject to K sampled global constraintsand the L local constraints, wherein the S users are sampled from the Nusers, and each of the K sampled global constraints restricts arespective scaled-down maximum total cost of resources across the Susers for each of the K pools.
 16. The non-transitory, computer-readablestorage medium of claim 11, wherein solving each of the N sub-problemscorresponding to the each of the N users comprises: for a sub-problemcorresponding to a user and for each dual multiplier corresponding tothe global constraint corresponding to the pool, computing candidates ofthe dual multiplier; sorting the candidates of the dual multiplier in anon-increasing order; traversing the candidates of the dual multiplierin the non-increasing order: for each candidate of the dual multiplier,computing M decision variables of the user corresponding to thecandidate of the dual multiplier; and computing an incremental cost ofselected resources based on the M decision variables of the usercorresponding to the candidate of the dual multiplier relative to a costof selected resources based on the M decision variables of the usercorresponding to a candidate of the dual multiplier preceding thecandidate of the dual multiplier in the non-increasing order; andwherein a total cost of resources selected across the N users for thepool corresponding to a certain dual multiplier is computed by summingincremental costs of selected resources of the user corresponding to allcandidates of the dual multiplier that are larger than or equal to thecertain dual multiplier.
 17. The non-transitory, computer-readablestorage medium of claim 16, wherein computing candidates of the dualmultiplier comprises computing pair-wise intersections points of Mlinear functions of the dual multiplier, wherein the M linear functionsof the dual multiplier represent M cost-adjusted rewards of the Mresources given the dual multiplier.
 18. The non-transitory,computer-readable storage medium of claim 16, the operations furthercomprising computing the non-negative threshold based on the candidatesof the dual multiplier sorted in the non-increasing order and arespective total cost of resources selected across the N users for thepool corresponding to one of the candidates of the dual multiplier inthe non-increasing order.
 19. The non-transitory, computer-readablestorage medium of claim 18, the operations further comprising performinginterpolation on the candidates of the dual multiplier sorted in thenon-increasing order for the computing the non-negative threshold. 20.The non-transitory, computer-readable storage medium of claim 16, theoperations further comprising: grouping the candidates of the dualmultiplier into a plurality of non-uniform budgets; computing sums ofincremental costs of selected resources of the user corresponding torespective candidates of the dual multiplier in the plurality ofnon-uniform budgets; and computing the non-negative threshold based onthe sums of incremental costs.
 21. A computer-implemented system,comprising: one or more computers; and one or more computer memorydevices interoperably coupled with the one or more computers and havingtangible, non-transitory, machine-readable media storing one or moreinstructions that, when executed by the one or more computers, performone or more operations for performing resource allocation, theoperations comprising: using a distributed computing system thatincludes a number of individual computer-implemented solvers forperforming resource allocation of M resources among N users into K poolsby solving a knapsack problem (KP) subject to K global constraints and Llocal constraints: receiving data representing the K global constraintsand the L local constraints, wherein each of the K global constraintsrestricts a respective maximum per-pool cost of the M resources acrosstwo or more users, and each of the L local constraints restricts aper-user selection of the M resources, transforming the KP into a dualproblem using K dual multipliers, each of the K dual multiplierscorresponding to a respective one of the K global constraints,decomposing the dual problem into N sub-problems, each of the Nsub-problems corresponding to a respective one of the N users andsubject to the L local constraints with respect to (w.r.t.) therespective one of the N users, performing two or more iterations insolving the dual problem, wherein in one iteration, for each dualmultiplier corresponding to a global constraint corresponding to a poolthat restricts a maximum per-pool cost of the M resources across two ormore users: determining an updated dual multiplier for the globalconstraint corresponding to the pool to be a non-negative threshold suchthat: a total cost of resources selected across the N users for the poolcorresponding to a dual multiplier that is less than the non-negativethreshold exceeds the maximum per-pool cost, and a total cost ofresources selected across the N users for the pool corresponding to adual multiplier that is not less than the non-negative threshold doesnot exceed the maximum per-pool cost, and computing M decision variablesof each of the N users corresponding to the updated dual multiplier insolving each of the N sub-problems corresponding to the each of the Nusers by one of the number of individual computer-implemented solvers,wherein each of the M decision variables indicates whether or not toselect a respective one of the M resources by the each of the N users.22. The system of claim 21, wherein N is in an order of billions orlarger, K is smaller than L, L is in an order of billions or larger. 23.The system of claim 21, wherein each of the L constraints corresponds toa subset of M resources and restricts a respective maximum number C ofresources to be selected among the subset of the M resources, whereinany subset corresponding to a first constraint out of the L constraintshas no common resource with any other subset corresponding to a secondconstraint out of the L constraints, unless the subset corresponding tothe first constraint is a subset of the other subset corresponding tothe second constraint.
 24. The system of claim 21, the operationsfurther comprising: in response to determining that a convergencecondition is met, computing M decision variables of each user of the Nusers corresponding to the K dual multipliers; and allocating the Mresources among the N users according to the M decision variables ofeach user of the N users.
 25. The system of claim 21, the operationsfurther comprising determining initial values of the K dual multipliers,wherein the initial values of the K dual multipliers are computed basedon a sampled version of the KP for resource allocation among S usersinto the K pools subject to K sampled global constraints and the L localconstraints, wherein the S users are sampled from the N users, and eachof the K sampled global constraints restricts a respective scaled-downmaximum total cost of resources across the S users for each of the Kpools.
 26. The system of claim 21, wherein solving each of the Nsub-problems corresponding to the each of the N users comprises: for asub-problem corresponding to a user and for each dual multipliercorresponding to the global constraint corresponding to the pool,computing candidates of the dual multiplier; sorting the candidates ofthe dual multiplier in a non-increasing order; traversing the candidatesof the dual multiplier in the non-increasing order: for each candidateof the dual multiplier, computing M decision variables of the usercorresponding to the candidate of the dual multiplier; and computing anincremental cost of selected resources based on the M decision variablesof the user corresponding to the candidate of the dual multiplierrelative to a cost of selected resources based on the M decisionvariables of the user corresponding to a candidate of the dualmultiplier preceding the candidate of the dual multiplier in thenon-increasing order; and wherein a total cost of resources selectedacross the N users for the pool corresponding to a certain dualmultiplier is computed by summing incremental costs of selectedresources of the user corresponding to all candidates of the dualmultiplier that are larger than or equal to the certain dual multiplier.27. The system of claim 26, wherein computing candidates of the dualmultiplier comprises computing pair-wise intersections points of Mlinear functions of the dual multiplier, wherein the M linear functionsof the dual multiplier represent M cost-adjusted rewards of the Mresources given the dual multiplier.
 28. The system of claim 26, theoperations further comprising computing the non-negative threshold basedon the candidates of the dual multiplier sorted in the non-increasingorder and a respective total cost of resources selected across the Nusers for the pool corresponding to one of the candidates of the dualmultiplier in the non-increasing order.
 29. The system of claim 28, theoperations further comprising performing interpolation on the candidatesof the dual multiplier sorted in the non-increasing order for thecomputing the non-negative threshold.
 30. The system of claim 26, theoperations further comprising: grouping the candidates of the dualmultiplier into a plurality of non-uniform budgets; computing sums ofincremental costs of selected resources of the user corresponding torespective candidates of the dual multiplier in the plurality ofnon-uniform budgets; and computing the non-negative threshold based onthe sums of incremental costs.